ńņš. 3 |

rationals of the form a/1, a ā Z. So a natural question is whether S ā’1 R contains a copy

of R.

2.8.3 Proposition

Deļ¬ne f : R ā’ S ā’1 R by f (a) = a/1. Then f is a ring homomorphism. If S has no zero

divisors then f is a monomorphism, and we say that R can be embedded in S ā’1 R. In

particular:

(i) A commutative ring R can be embedded in its complete (or full ) ring of fractions

(S ā’1 R, where S consists of all non-divisors of zero in R).

(ii) An integral domain can be embedded in its quotient ļ¬eld.

Proof. We have f (a + b) = a+b = a + 1 = f (a) + f (b), f (ab) = ab = a 1 = f (a)f (b),

b b

1 1 1 1

and f (1) = 1 , proving that f is a ring homomorphism. If S has no zero divisors and

1

f (a) = a/1 = 0/1, then for some s ā S we have sa = 0, and since s cannot be a zero

divisor, we have a = 0. Thus f is a monomorphism. ā™£

2.8.4 Corollary

The quotient ļ¬eld F of an integral domain R is the smallest ļ¬eld containing R.

Proof. By (2.8.3), we may regard R as a subset of F , so that F is a ļ¬eld containing R.

But if L is any ļ¬eld containing R, then all fractions a/b, a, b ā R, must belong to L. Thus

F ā L. ā™£

30 CHAPTER 2. RING FUNDAMENTALS

Problems For Section 2.8

1. If the integral domain D is in fact a ļ¬eld, what is the quotient ļ¬eld of D?

2. If D is the set F [X] of all polynomials over a ļ¬eld F , what is the quotient ļ¬eld of D?

3. Give a detailed proof that addition in a ring of fractions is associative.

4. Give a detailed proof that the distributive laws hold in a ring of fractions.

5. Let R be an integral domain with quotient ļ¬eld F , and let h be a ring monomorphism

from R to a ļ¬eld L. Show that h has a unique extension to a monomorphism from F

to L.

6. Let h be the ring homomorphism from Z to Zp , p prime, given by h(x) = x mod p. Why

canā™t the analysis of Problem 5 be used to show that h extends to a monomorphism of

the rationals to Zp ? (This canā™t possibly work since Zp is ļ¬nite, but what goes wrong?)

7. Let S be a multiplicative subset of the commutative ring R, with f : R ā’ S ā’1 R deļ¬ned

by f (a) = a/1. If g is a ring homomorphism from R to a commutative ring R and g(s)

is a unit in R for each s ā S, we wish to ļ¬nd a ring homomorphism g : S ā’1 R ā’ R such

that g(f (a)) = g(a) for every a ā R, i.e., such that the diagram below is commutative.

Show that there is only one conceivable way to deļ¬ne g.

GR

g

R

x`

xx

xx

x

f

xx g

S ā’1 R

8. Show that the mapping you have deļ¬ned in Problem 7 is a well-deļ¬ned ring homomor-

phism.

2.9 Irreducible Polynomials

2.9.1 Deļ¬nitions and Comments

In (2.6.1) we deļ¬ned an irreducible element of a ring; it is a nonzero nonunit which cannot

be represented as a product of nonunits. If R is an integral domain, we will refer to an

irreducible element of R[X] as an irreducible polynomial. Now in F [X], where F is a ļ¬eld,

the units are simply the nonzero elements of F (Section 2.1, Problem 2). Thus in this

case, an irreducible element is a polynomial of degree at least 1 that cannot be factored

into two polynomials of lower degree. A polynomial that is not irreducible is said to be

reducible or factorable. For example, X 2 + 1, regarded as an element of R[X], where R

is the ļ¬eld of real numbers, is irreducible, but if we replace R by the larger ļ¬eld C of

ā

complex numbers, X 2 + 1 is factorable as (X ā’ i)(X + i), i = ā’1. We say that X 2 + 1

is irreducible over R but not irreducible over C.

Now consider D[X], where D is a unique factorization domain but not necessarily

a ļ¬eld, for example, D = Z. The polynomial 12X + 18 is not an irreducible element

of Z[X] because it can be factored as the product of the two nonunits 6 and 2X + 3.

2.9. IRREDUCIBLE POLYNOMIALS 31

It is convenient to factor out the greatest common divisor of the coeļ¬cients (6 in this

case). The result is a primitive polynomial, one whose content (gcd of coeļ¬cients) is 1.

A primitive polynomial will be irreducible if and only if it cannot be factored into two

polynomials of lower degree.

In this section, we will compare irreducibility over a unique factorization domain D

and irreducibility over the quotient ļ¬eld F of D. Here is the key result.

2.9.2 Proposition

Let D be a unique factorization domain with quotient ļ¬eld F . Suppose that f is a nonzero

polynomial in D[X] and that f can be factored as gh, where g and h belong to F [X].

Then there is a nonzero element Ī» ā F such that Ī»g ā D[X] and Ī»ā’1 h ā D[X]. Thus if f

is factorable over F , then it is factorable over D. Equivalently, if f is irreducible over D,

then f is irreducible over F .

Proof. The coeļ¬cients of g and h are quotients of elements of D. If a is the least common

denominator for g (technically, the least common multiple of the denominators of the

coeļ¬cients of g), let g ā— = ag ā D[X]. Similarly, let hā— = bh ā D[X]. Thus abf = g ā— hā—

with g ā— , hā— ā D[X] and c = ab ā D.

Now if p is a prime factor of c, we will show that either p divides all coeļ¬cients of g ā—

or p divides all coeļ¬cients of hā— . We do this for all prime factors of c to get f = g0 h0

with g0 , h0 ā D[X]. Since going from g to g0 involves only multiplication or division by

nonzero constants in D, we have g0 = Ī»g for some nonzero Ī» ā F . But then h0 = Ī»ā’1 h,

as desired.

Now let

g ā— (X) = g0 + g1 X + Ā· Ā· Ā· + gs X s , hā— (X) = h0 + h1 X + Ā· Ā· Ā· + ht X t .

Since p is a prime factor of c = ab and abf = g ā— hā— , p must divide all coeļ¬cients of g ā— hā— .

If p does not divide every gi and p does not divide every hi , let gu and hv be the coeļ¬cients

of minimum index not divisible by p. Then the coeļ¬cient of X u+v in g ā— hā— is

g0 hu+v + g1 hu+vā’1 + Ā· Ā· Ā· + gu hv + Ā· Ā· Ā· + gu+vā’1 h1 + gu+v h0 .

But by choice of u and v, p divides every term of this expression except gu hv , so that p

cannot divide the entire expression. So there is a coeļ¬cient of g ā— hā— not divisible by p, a

contradiction. ā™£

The technique of the above proof yields the following result.

2.9.3 Gaussā™ Lemma

Let f and g be nonconstant polynomials in D[X], where D is a unique factorization

domain. If c denotes content, then c(f g) = c(f )c(g), up to associates. In particular, the

product of two primitive polynomials is primitive.

32 CHAPTER 2. RING FUNDAMENTALS

Proof. By deļ¬nition of content we may write f = c(f )f ā— and g = c(g)g ā— where f ā— and g ā—

are primitive. Thus f g = c(f )c(g)f ā— g ā— . It follows that c(f )c(g) divides every coeļ¬cient

of f g, so c(f )c(g) divides c(f g). Now let p be any prime factor of c(f g); then p divides

c(f )c(g)f ā— g ā— , and the proof of (2.9.2) shows that either p divides every coeļ¬cient of

c(f )f ā— or p divides every coeļ¬cient of c(g)g ā— . If, say, p divides every coeļ¬cient of c(f )f ā— ,

then (since p is prime) either p divides c(f ) or p divides every coeļ¬cient of f ā— . But f ā— is

primitive, so that p divides c(f ), hence p divides c(f )c(g). We conclude that c(f g) divides

c(f )c(g), and the result follows. ā™£

2.9.4 Corollary of the Proof of (2.9.3)

If h is a nonconstant polynomial in D[X] and h = ahā— where hā— is primitive and a ā D,

then a must be the content of h.

Proof. Since a divides every coeļ¬cient of h, a must divide c(h). If p is any prime factor

of c(h), then p divides every coeļ¬cient of ahā— , and as in (2.9.3), either p divides a or

p divides every coeļ¬cient of hā— , which is impossible by primitivity of hā— . Thus c(h)

divides a, and the result follows. ā™£

Proposition 2.9.2 yields a precise statement comparing irreducibility over D with ir-

reducibility over F .

2.9.5 Proposition

Let D be a unique factorization domain with quotient ļ¬eld F . If f is a nonconstant

polynomial in D[X], then f is irreducible over D if and only if f is primitive and irreducible

over F .

Proof. If f is irreducible over D, then f is irreducible over F by (2.9.2). If f is not

primitive, then f = c(f )f ā— where f ā— is primitive and c(f ) is not a unit. This contradicts

the irreducibility of f over D. Conversely, if f = gh is a factorization of the primitive

polynomial f over D, then g and h must be of degree at least 1. Thus neither g nor h is

a unit in F [X], so f = gh is a factorization of f over F . ā™£

Here is another basic application of (2.9.2).

2.9.6 Theorem

If R is a unique factorization domain, so is R[X].

Proof. If f ā R[X], f = 0, then f can be factored over the quotient ļ¬eld F as f =

f1 f2 . . . fk , where the fi are irreducible polynomials in F [X]. (Recall that F [X] is a

Euclidean domain, hence a unique factorization domain.) By (2.9.2), for some nonzero

Ī»1 ā F we may write f = (Ī»1 f1 )(Ī»ā’1 f2 . . . fk ) with Ī»1 f1 and Ī»ā’1 f2 . . . fk in R[X]. Again

1 1

by (2.9.2), we have

Ī»ā’1 f2 . . . fk = f2 Ī»ā’1 f3 . . . fk = (Ī»2 f2 )(Ī»ā’1 Ī»ā’1 f3 . . . fk )

1 1 2 1

2.9. IRREDUCIBLE POLYNOMIALS 33

with Ī»2 f2 and Ī»ā’1 Ī»ā’1 f3 . . . fk ā R[X]. Continuing inductively, we express f as i=1 Ī»i fi

k

2 1

where the Ī»i fi are in R[X] and are irreducible over F . But Ī»i fi is the product of its

content and a primitive polynomial (which is irreducible over F , hence over R by (2.9.5)).

Furthermore, the content is either a unit or a product of irreducible elements of the UFD

R, and these elements are irreducible in R[X] as well. This establishes the existence of a

factorization into irreducibles.

Now suppose that f = g1 Ā· Ā· Ā· gr = h1 Ā· Ā· Ā· hs , where the gi and hi are nonconstant

irreducible polynomials in R[X]. (Constant polynomials cause no diļ¬culty because R is

a UFD.) By (2.9.5), the gi and hi are irreducible over F , and since F [X] is a UFD, we

have r = s and, after reordering if necessary, gi and hi are associates (in F [X]) for each

i. Now gi = ci hi for some constant ci ā F , and we have ci = ai /bi with ai , bi ā R. Thus

bi gi = ai hi , with gi and hi primitive by (2.9.5). By (2.9.4), bi gi has content bi and ai hi

has content ai . Therefore ai and bi are associates, which makes ci a unit in R, which in

turn makes gi and hi associates in R[X], proving uniqueness of factorization. ā™£

The following result is often used to establish irreducibility of a polynomial.

2.9.7 Eisensteinā™s Irreducibility Criterion

Let R be a UFD with quotient ļ¬eld F , and let f (X) = an X n + Ā· Ā· Ā· + a1 X + a0 be a

polynomial in R[X], with n ā„ 1 and an = 0. If p is prime in R and p divides ai for

0 ā¤ i < n, but p does not divide an and p2 does not divide a0 , then f is irreducible over

F . Thus by (2.9.5), if f is primitive then f is irreducible over R.

Proof. If we divide f by its content to produce a primitive polynomial f ā— , the hypothesis

still holds for f ā— . (Since p does not divide an , it is not a prime factor of c(f ), so it must

divide the ith coeļ¬cient of f ā— for 0 ā¤ i < n.) If we can prove that f ā— is irreducible

over R, then by (2.9.5), f ā— is irreducible over F , and therefore so is f . Thus we may

assume without loss of generality that f is primitive, and prove that f is irreducible

over R.

Assume that f = gh, with g(X) = g0 +Ā· Ā· Ā·+gr X r and h(X) = h0 +Ā· Ā· Ā·+hs X s . If r = 0

then g0 divides all coeļ¬cients ai of f , so g0 divides c(f ), hence g(= g0 ) is a unit. Thus we

may assume that r ā„ 1, and similarly s ā„ 1. By hypothesis, p divides a0 = g0 h0 but p2

does not divide a0 , so p cannot divide both g0 and h0 . Assume that p fails to divide h0 ,

so that p divides g0 ; the argument is symmetrical in the other case. Now gr hs = an , and

by hypothesis, p does not divide an , so that p does not divide gr . Let i be the smallest

integer such that p does not divide gi ; then 1 ā¤ i ā¤ r < n (since r + s = n and s ā„ 1).

Now

ai = g0 hi + g1 hiā’1 + Ā· Ā· Ā· + gi h0

and by choice of i, p divides g0 , . . . , giā’1 . But p divides the entire sum ai , so p must divide

the last term gi h0 . Consequently, either p divides gi , which contradicts the choice of i, or

p divides h0 , which contradicts our earlier assumption. Thus there can be no factorization

of f as a product of polynomials of lower degree; in other words, f is irreducible over R. ā™£

34 CHAPTER 2. RING FUNDAMENTALS

Problems For Section 2.9

1. (The rational root test, which can be useful in factoring a polynomial over Q.)

Let f (X) = an X n + Ā· Ā· Ā· + a1 X + a0 ā Z[X]. If f has a rational root u/v where u and v

are relatively prime integers and v = 0, show that v divides an and u divides a0 .

2. Show that for every positive integer n, there is at least one irreducible polynomial of

degree n over the integers.

3. If f (X) ā Z[X] and p is prime, we can reduce all coeļ¬cients of f modulo p to obtain

a new polynomial fp (X) ā Zp [X]. If f is factorable over Z, then fp is factorable over

Zp . Therefore if fp is irreducible over Zp , then f is irreducible over Z. Use this idea

to show that the polynomial X 3 + 27X 2 + 5X + 97 is irreducible over Z. (Note that

Eisenstein does not apply.)

4. If we make a change of variable X = Y + c in the polynomial f (X), the result is a new

polynomial g(Y ) = f (Y + c). If g is factorable over Z, so is f since f (X) = g(X ā’ c).

Thus if f is irreducible over Z, so is g. Use this idea to show that X 4 +4X 3 +6X 2 +4X+4

is irreducible over Z.

5. Show that in Z[X], the ideal n, X , n ā„ 2, is not principal, and therefore Z[X] is a

UFD that is not a PID.

aij X i Y j , aij ā F ,

6. Show that if F is a ļ¬eld, then F [X, Y ], the set of all polynomials

is not a PID since the ideal X, Y is not principal.

7. Let f (X, Y ) = X 2 + Y 2 + 1 ā C[X, Y ], where C is the ļ¬eld of complex numbers.

Write f as Y 2 + (X 2 + 1) and use Eisensteinā™s criterion to show that f is irreducible

over C.

8. Show that f (X, Y ) = X 3 + Y 3 + 1 is irreducible over C.

Chapter 3

Field Fundamentals

3.1 Field Extensions

If F is a ļ¬eld and F [X] is the set of all polynomials over F, that is, polynomials with

coeļ¬cients in F , we know that F [X] is a Euclidean domain, and therefore a principal ideal

domain and a unique factorization domain (see Sections 2.6 and 2.7). Thus any nonzero

polynomial f in F [X] can be factored uniquely as a product of irreducible polynomials.

Any root of f must be a root of one of the irreducible factors, but at this point we have

no concrete information about the existence of roots and how they might be found. For

example, X 2 + 1 has no real roots, but if we consider the larger ļ¬eld of complex numbers,

we get two roots, +i and ā’i. It appears that the process of passing to a larger ļ¬eld may

help produce roots, and this turns out to be correct.

3.1.1 Deļ¬nitions

If F and E are ļ¬elds and F ā E, we say that E is an extension of F , and we write F ā¤ E,

or sometimes E/F .

If E is an extension of F , then in particular E is an abelian group under addition, and

we may multiply the āvectorā x ā E by the āscalarā Ī» ā F , and the axioms of a vector

space are satisļ¬ed. Thus if F ā¤ E, then E is a vector space over F . The dimension of this

vector space is called the degree of the extension, written [E : F ]. If [E : F ] = n < ā, we

say that E is a ļ¬nite extension of F , or that the extension E/F is ļ¬nite, or that E is of

degree n over F .

If f is a nonconstant polynomial over the ļ¬eld F , and f has no roots in F , we can

always produce a root of f in an extension ļ¬eld of F . We do this after a preliminary

result.

3.1.2 Lemma

Let f : F ā’ E be a homomorphism of ļ¬elds, i.e., f (a + b) = f (a) + f (b), f (ab) = f (a)f (b)

(all a, b ā F ), and f (1F ) = 1E . Then f is a monomorphism.

1

2 CHAPTER 3. FIELD FUNDAMENTALS

Proof. First note that a ļ¬eld F has no ideals except {0} and F . For if a is a nonzero

member of the ideal I, then ab = 1 for some b ā F , hence 1 ā I, and therefore I = F .

Taking I to be the kernel of f , we see that I cannot be all of F because f (1) = 0. Thus

I must be {0}, so that f is injective. ā™£

3.1.3 Theorem

Let f be a nonconstant polynomial over the ļ¬eld F . Then there is an extension E/F and

an element Ī± ā E such that f (Ī±) = 0.

Proof. Since f can be factored into irreducibles, we may assume without loss of generality

that f itself is irreducible. The ideal I = f (X) in F [X] is prime (see (2.6.1)), in fact

maximal (see (2.6.9)). Thus E = F [X]/I is a ļ¬eld by (2.4.3). We have a problem at this

point because F need not be a subset of E, but we can place an isomorphic copy of F

inside E via the homomorphism h : a ā’ a + I; by (3.1.2), h is a monomorphism, so we

may identify F with a subļ¬eld of E. Now let Ī± = X + I; if f (X) = a0 + a1 X + Ā· Ā· Ā· + an X n ,

then

f (Ī±) = (a0 + I) + a1 (X + I) + Ā· Ā· Ā· + an (X + I)n

= (a0 + a1 X + Ā· Ā· Ā· + an X n ) + I

= f (X) + I

which is zero in E. ā™£

The extension E is sometimes said to be obtained from F by adjoining a root Ī± of f .

Here is a further connection between roots and extensions.

3.1.4 Proposition

Let f and g be polynomials over the ļ¬eld F . Then f and g are relatively prime if and

only if f and g have no common root in any extension of F .

Proof. If f and g are relatively prime, their greatest common divisor is 1, so there are

polynomials a(X) and b(X) over F such that a(X)f (X) + b(X)g(X) = 1. If Ī± is a

common root of f and g, then the substitution of Ī± for X yields 0 = 1, a contradiction.

Conversely, if the greatest common divisor d(X) of f (X) and g(X) is nonconstant, let E

be an extension of F in which d(X) has a root Ī± (E exists by (3.1.3)). Since d(X) divides

both f (X) and g(X), Ī± is a common root of f and g in E. ā™£

3.1.5 Corollary

If f and g are distinct monic irreducible polynomials over F , then f and g have no common

roots in any extension of F .

Proof. If h is a nonconstant divisor of the irreducible polynomials f and g, then up

to multiplication by constants, h coincides with both f and g, so that f is a constant

multiple of g. This is impossible because f and g are monic and distinct. Thus f and g

are relatively prime, and the result follows from (3.1.4). ā™£

3.1. FIELD EXTENSIONS 3

If E is an extension of F and Ī± ā E is a root of a polynomial f ā F [X], it is often

of interest to examine the ļ¬eld F (Ī±) generated by F and Ī±, in other words the smallest

subļ¬eld of E containing F and Ī± (more precisely, containing all elements of F along

with Ī±). The ļ¬eld F (Ī±) can be described abstractly as the intersection of all subļ¬elds of

E containing F and Ī±, and more concretely as the collection of all rational functions

a0 + a1 Ī± + Ā· Ā· Ā· + am Ī±m

b0 + b1 Ī± + Ā· Ā· Ā· + b n Ī± n

with ai , bj ā F, m, n = 0, 1, . . . , and b0 + b1 Ī± + Ā· Ā· Ā· + bn Ī±n = 0. In fact there is a much

less complicated description of F (Ī±), as we will see shortly.

3.1.6 Deļ¬nitions and Comments

If E is an extension of F , the element Ī± ā E is said to be algebraic over F is there is a

nonconstant polynomial f ā F [X] such that f (Ī±) = 0; if Ī± is not algebraic over F , it is

said to be transcendental over F . If every element of E is algebraic over F , then E is said

to be an algebraic extension of F .

Suppose that Ī± ā E is algebraic over F , and let I be the set of all polynomials g

over F such that g(Ī±) = 0. If g1 and g2 belong to I, so does g1 Ā± g2 , and if g ā I and

c ā F [X], then cg ā I. Thus I is an ideal of F [X], and since F [X] is a PID, I consists of

all multiples of some m(X) ā F [X]. Any two such generators must be multiples of each

other, so if we require that m(X) be monic, then m(X) is unique. The polynomial m(X)

has the following properties:

(1) If g ā F [X], then g(Ī±) = 0 if and only if m(X) divides g(X).

(2) m(X) is the monic polynomial of least degree such that m(Ī±) = 0.

(3) m(X) is the unique monic irreducible polynomial such that m(Ī±) = 0.

Property (1) follows because g(Ī±) = 0 iļ¬ g(X) ā I, and I = m(X) , the ideal generated

by m(X). Property (2) follows from (1). To prove (3), note that if m(X) = h(X)k(X)

with deg h and deg k less than deg m, then either h(Ī±) = 0 or k(Ī±) = 0, so that by (1),

either h(X) or k(X) is a multiple of m(X), which is impossible. Thus m(X) is irreducible,

and uniqueness of m(X) follows from (3.1.5).

The polynomial m(X) is called the minimal polynomial of Ī± over F , sometimes written

as min(Ī±, F ).

3.1.7 Theorem

If Ī± ā E is algebraic over F and the minimal polynomial m(X) of Ī± over F has degree

n, then F (Ī±) = F [Ī±], the set of polynomials in Ī± with coeļ¬cients in F . In fact, F [Ī±]

is the set Fnā’1 [Ī±] of all polynomials of degree at most n ā’ 1 with coeļ¬cients in F , and

1, Ī±, . . . , Ī±nā’1 form a basis for the vector space F [Ī±] over the ļ¬eld F . Consequently,

[F (Ī±) : F ] = n.

Proof. Let f (X) be any nonzero polynomial over F of degree n ā’ 1 or less. Then since

m(X) is irreducible and deg f < deg m, f (X) and m(X) are relatively prime, and there

4 CHAPTER 3. FIELD FUNDAMENTALS

are polynomials a(X) and b(X) over F such that a(X)f (X) + b(X)m(X) = 1. But

then a(Ī±)f (Ī±) = 1, so that any nonzero element of Fnā’1 [Ī±] has a multiplicative inverse.

It follows that Fnā’1 [Ī±] is a ļ¬eld. (This may not be obvious, since the product of two

polynomials of degree n ā’ 1 or less can have degree greater than n ā’ 1, but if deg g > n ā’ 1,

then divide g by m to get g(X) = q(X)m(X) + r(X) where deg r(X) < deg m(X) = n.

Replace X by Ī± to get g(Ī±) = r(Ī±) ā Fnā’1 [Ī±]. Less abstractly, if m(Ī±) = Ī±3 + Ī± + 1 = 0,

then Ī±3 = ā’Ī± ā’ 1, Ī±4 = ā’Ī±2 ā’ Ī±, and so on.)

Now any ļ¬eld containing F and Ī± must contain all polynomials in Ī±, in particular

all polynomials of degree at most n ā’ 1. Therefore Fnā’1 [Ī±] ā F [Ī±] ā F (Ī±). But F (Ī±)

is the smallest ļ¬eld containing F and Ī±, so F (Ī±) ā Fnā’1 [Ī±], and we conclude that

F (Ī±) = F [Ī±] = Fnā’1 [Ī±]. Finally, the elements 1, Ī±, . . . , Ī±nā’1 certainly span Fnā’1 [Ī±], and

they are linearly independent because if a nontrivial linear combination of these elements

were zero, we would have a nonzero polynomial of degree less than that of m(X) with Ī±

as a root, contradicting (2) of (3.1.6). ā™£

We now prove a basic multiplicativity result for extensions, after a preliminary dis-

cussion.

3.1.8 Lemma

Suppose that F ā¤ K ā¤ E, the elements Ī±i , i ā I, form a basis for E over K, and the

elements Ī²j , j ā J, form a basis for K over F . (I and J need not be ļ¬nite.) Then the

products Ī±i Ī²j , i ā I, j ā J, form a basis for E over F .

Proof. If Ī³ ā E, then Ī³ is a linear combination of the Ī±i with coeļ¬cients ai ā K, and

each ai is a linear combination of the Ī²j with coeļ¬cients bij ā F . It follows that the Ī±i Ī²j

span E over F . Now if i,j Ī»ij Ī±i Ī²j = 0, then i Ī»ij Ī±i = 0 for all j, and consequently

Ī»ij = 0 for all i, j, and the Ī±i Ī²j are linearly independent. ā™£

3.1.9 The Degree is Multiplicative

If F ā¤ K ā¤ E, then [E : F ] = [E : K][K : F ]. In particular, [E : F ] is ļ¬nite if and only if

[E : K] and [K : F ] are both ļ¬nite.

Proof. In (3.1.8), we have [E : K] = |I|, [K : F ] = |J|, and [E : F ] = |I||J|. ā™£

We close this section by proving that every ļ¬nite extension is algebraic.

3.1.10 Theorem

If E is a ļ¬nite extension of F , then E is an algebraic extension of F .

Proof. Let Ī± ā E, and let n = [E : F ]. Then 1, Ī±, Ī±2 , . . . , Ī±n are n + 1 vectors in an

n-dimensional vector space, so they must be linearly dependent. Thus Ī± is a root of a

nonzero polynomial with coeļ¬cients in F , which means that Ī± is algebraic over F . ā™£

3.2. SPLITTING FIELDS 5

Problems For Section 3.1

1. Let E be an extension of F , and let S be a subset of E. If F (S) is the subļ¬eld of E

generated by S over F , in other words, the smallest subļ¬eld of E containing F and S,

describe F (S) explicitly, and justify your characterization.

2. If for each i ā I, Ki is a subļ¬eld of the ļ¬eld E, the composite of the Ki (notation

i Ki ) is the smallest subļ¬eld of E containing every Ki . As in Problem 1, describe

the composite explicitly.

3. Assume that Ī± is algebraic over F , with [F [Ī±] : F ] = n. If Ī² ā F [Ī±], show that

[F [Ī²] : F ] ā¤ n, in fact [F [Ī²] : F ] divides n.

ā

4. The minimal polynomial of 2 over the rationals Q is X 2 ā’ 2, by (3) of (3.1.6). Thus

ā ā

Q[ 2] consists of all numbers of the form a0 + a1 2, where a0 and a1 are rational.

ā

By Problem 3, we know that ā’1 + 2 has a minimal polynomial over Q of degree at

most 2. Find this minimal polynomial.

5. If Ī± is algebraic over F and Ī² belongs to F [Ī±], describe a systematic procedure for

ļ¬nding the minimal polynomial of Ī² over F .

6. If E/F and the element Ī± ā E is transcendental over F , show that F (Ī±) is isomorphic

to F (X), the ļ¬eld of rational functions with coeļ¬cients in F .

7. Theorem 3.1.3 gives one method of adjoining a root of a polynomial, and in fact there

is essentially only one way to do this. If E is an extension of F and Ī± ā E is algebraic

over F with minimal polynomial m(X), let I be the ideal m(X) ā F [X]. Show that

F (Ī±) is isomorphic to F [X]/I. [Deļ¬ne Ļ• : F [X] ā’ E by Ļ•(f (X)) = f (Ī±), and use

the ļ¬rst isomorphism theorem for rings.]

8. In the proof of (3.1.3), we showed that if f is irreducible in F [X], then I = f is a

maximal ideal. Show that conversely, if I is a maximal ideal, then f is irreducible.

9. Suppose that F ā¤ E ā¤ L, with Ī± ā L. What is the relation between the minimal

polynomial of Ī± over F and the minimal polynomial of Ī± over E?

10. If Ī±1 , . . . , Ī±n are algebraic over F , we can successively adjoin the Ī±i to F to obtain

the ļ¬eld F [Ī±1 , . . . , Ī±n ] consisting of all polynomials over F in the Ī±i . Show that

n

[F [Ī±1 , . . . , Ī±n ] : F ] ā¤ [F (Ī±i ) : F ] < ā

i=1

3.2 Splitting Fields

If f is a polynomial over the ļ¬eld F , then by (3.1.3) we can ļ¬nd an extension E1 of F

containing a root Ī±1 of f . If not all roots of f lie in E1 , we can ļ¬nd an extension E2

of E1 containing another root Ī±2 of f . If we continue the process, eventually we reach a

complete factorization of f . In this section we examine this idea in detail.

If E is an extension of F and Ī±1 , . . . , Ī±k ā E, we will use the notation F (Ī±1 , . . . , Ī±k )

for the subļ¬eld of E generated by F and the Ī±i . Thus F (Ī±1 , . . . , Ī±k ) is the smallest

subļ¬eld of E containing all elements of F along with the Ī±i . ( āSmallestā means that

F (Ī±1 , . . . , Ī±k ) is the intersection of all such subļ¬elds.) Explicitly, F (Ī±1 , . . . , Ī±k ) is the

collection of all rational functions in the Ī±i with nonzero denominators.

6 CHAPTER 3. FIELD FUNDAMENTALS

3.2.1 Deļ¬nitions and Comments

If E is an extension of F and f ā F [X], we say that f splits over E if f can be written

as Ī»(X ā’ Ī±1 ) Ā· Ā· Ā· (X ā’ Ī±k ) for some Ī±1 , . . . , Ī±k ā E and Ī» ā F .

(There is a subtle point that should be mentioned. We would like to refer to the Ī±i

as ātheā roots of f , but in doing so we are implicitly assuming that if Ī² is an element of

some extension E of E and f (Ī²) = 0, then Ī² must be one of the Ī±i . This follows upon

substituting Ī² into the equation f (X) = Ī»(X ā’ Ī±1 ) Ā· Ā· Ā· (X ā’ Ī±k ) = 0.)

If K is an extension of F and f ā F [X], we say that K is a splitting ļ¬eld for f over

F if f splits over K but not over any proper subļ¬eld of K containing F .

Equivalently, K is a splitting ļ¬eld for f over F if f splits over K and K is generated

over F by the roots Ī±1 , . . . , Ī±k of f , in other words, F (Ī±1 , . . . , Ī±k ) = K. For if K is a

splitting ļ¬eld for f , then since f splits over K we have all Ī±j ā K, so F (Ī±1 , . . . , Ī±k ) ā K.

But f splits over F (Ī±1 , . . . , Ī±k ), and it follows that F (Ī±1 , . . . , Ī±k ) cannot be a proper

subļ¬eld; it must coincide with K. Conversely, if f splits over K and F (Ī±1 , . . . , Ī±k ) = K,

let L be a subļ¬eld of K containing F . If f splits over L then all Ī±i belong to L, so

K = F (Ī±1 , . . . , Ī±k ) ā L ā K, so L = K.

If f ā F [X] and f splits over the extension E of F , then E contains a unique splitting

ļ¬eld for f , namely F (Ī±1 , . . . , Ī±k ).

3.2.2 Proposition

If f ā F [X] and deg f = n, then f has a splitting ļ¬eld K over F with [K : F ] ā¤ n!.

Proof. We may assume that n ā„ 1. (If f is constant, take K = F .) By (3.1.3), F has an

extension E1 containing a root Ī±1 of f , and the extension F (Ī±1 )/F has degree at most

n. (Since f (Ī±1 ) = 0, the minimal polynomial of Ī±1 divides f ; see (3.1.6) and (3.1.7).) We

may then write f (X) = (X ā’ Ī±1 )r1 g(X), where Ī±1 is not a root of g and deg g ā¤ n ā’ 1.

If g is nonconstant, we can ļ¬nd an extension of F (Ī±1 ) containing a root Ī±2 of g, and the

extension F (Ī±1 , Ī±2 ) will have degree at most n ā’ 1 over F (Ī±1 ). Continue inductively and

use (3.1.9) to reach an extension of degree at most n! containing all the roots of f . ā™£

If f ā F [X] and f splits over E, then we may pick any root Ī± of f and adjoin it to F

to obtain the extension F (Ī±). Roots of the same irreducible factor of f yield essentially

the same extension, as the next result shows.

3.2.3 Theorem

If Ī± and Ī² are roots of the irreducible polynomial f ā F [X] in an extension E of F , then

F (Ī±) is isomorphic to F (Ī²) via an isomorphism that carries Ī± into Ī² and is the identity

on F .

Proof. Without loss of generality we may assume f monic (if not, divide f by its leading

coeļ¬cient). By (3.1.6), part (3), f is the minimal polynomial of both Ī± and Ī². By (3.1.7),

the elements of F (Ī±) can be expressed uniquely as a0 + a1 Ī± + Ā· Ā· Ā· + anā’1 Ī±nā’1 , where the

ai belong to F and n is the degree of f . The desired isomorphism is given by

a0 + a1 Ī± + Ā· Ā· Ā· + anā’1 Ī±nā’1 ā’ a0 + a1 Ī² + Ā· Ā· Ā· + anā’1 Ī² nā’1 . ā™£

3.2. SPLITTING FIELDS 7

If f is a polynomial in F [X] and F is isomorphic to the ļ¬eld F via the isomorphism i,

we may regard f as a polynomial over F . We simply use i to transfer f . Thus if

f = a0 + a1 X + Ā· Ā· Ā· an X n , then f = i(f ) = i(a0 ) + i(a1 )X + Ā· Ā· Ā· + i(an )X n . There is only

a notational diļ¬erence between f and f , and we expect that splitting ļ¬elds for f and f

should also be essentially the same. We prove this after the following deļ¬nition.

3.2.4 Deļ¬nition

If E and E are extensions of F and i is an isomorphism of E and E , we say that i is

an F -isomorphism if i ļ¬xes F , that is, if i(a) = a for every a ā F . F -homomorphisms,

F -monomorphisms, etc., are deļ¬ned similarly.

3.2.5 Isomorphism Extension Theorem

Suppose that F and F are isomorphic, and the isomorphism i carries the polynomial

f ā F [X] to f ā F [X]. If K is a splitting ļ¬eld for f over F and K is a splitting ļ¬eld

for f over F , then i can be extended to an isomorphism of K and K . In particular, if

F = F and i is the identity function, we conclude that any two splitting ļ¬elds of f are

F -isomorphic.

Proof. Carry out the construction of a splitting ļ¬eld for f over F as in (3.2.2), and perform

exactly the same steps to construct a splitting ļ¬eld for f over F . At every stage, there is

only a notational diļ¬erence between the ļ¬elds obtained. Furthermore, we can do the ļ¬rst

construction inside K and the second inside K . But the comment at the end of (3.2.1)

shows that the splitting ļ¬elds that we have constructed coincide with K and K . ā™£

3.2.6 Example

We will ļ¬nd a splitting ļ¬eld for f (X) = X 3 ā’ 2 over the rationals Q. ā

If Ī± is the positive cube root of 2, then the roots of f are Ī±, Ī±(ā’ 1 + i 1 3) and

ā 2 2

Ī±(ā’ 2 ā’ i 2 3). The polynomial f is irreducible, either by Eisensteinā™s criterion or by the

1 1

observation that if f were factorable, it would have a linear factor, and there is no rational

number whose cube is 2. Thus f is the minimal polynomial of Ī±, so [Q(Ī±) : Q] ā 3. Now

=

ā

since Ī± and i 3 generate all the roots of f , the splitting ļ¬eld is K = Q(Ī±, i 3). (We

ā

complex numbers C.) Since i 3 ā Q(Ī±)

ā

regard all ļ¬elds in this example as subļ¬elds of the ā /

(because Q(Ī±) is a subļ¬eld of the reals), [Q(Ī±, iā 3) : Q(Ī±)] is at least 2. But i 3 is a

root of X 2 + 3 ā Q(Ī±)[X], so the degree of Q(Ī±, i 3) over Q(Ī±) is a most 2, and therefore

is exactly 2. Thus

ā ā

[K : Q] = [Q(Ī±, i 3) : Q] = [Q(Ī±, i 3) : Q(Ī±)][Q(Ī±) : Q] = 2 Ć— 3 = 6.

Problems For Section 3.2

1. Find a splitting ļ¬eld for f (X) = X 2 ā’ 4X + 4 over Q.

2. Find a splitting ļ¬eld K for f (X) = X 2 ā’ 2X + 4 over Q, and determine the degree of

K over Q.

8 CHAPTER 3. FIELD FUNDAMENTALS

3. Find a splitting ļ¬eld K for f (X) = X 4 ā’ 2 over Q, and determine [K : Q].

4. Let C be a family of polynomials over F , and let K be an extension of F . Show that

the following two conditions are equivalent:

(a) Each f ā C splits over K, but if F ā¤ K < K, then it is not true that each f ā C

splits over K .

(b) Each f ā C splits over K, and K is generated over F by the roots of all the

polynomials in C.

If one, and hence both, of these conditions are satisļ¬ed, we say that K is a splitting

ļ¬eld for C over F .

5. Suppose that K is a splitting ļ¬eld for the ļ¬nite set of polynomials {f1 , . . . , fr } over F .

Express K as a splitting ļ¬eld for a single polynomial f over F .

6. If m and n are ā ā square-free positive integers greater than 1, show that the

distinct

splitting ļ¬eld Q( m, n) of (X 2 ā’ m)(X 2 ā’ n) has degree 4 over Q.

3.3 Algebraic Closures

If f is a polynomial of degree n over the rationals or the reals, or more generally over

the complex numbers, then f need not have any rational roots, or even real roots, but we

know that f always has n complex roots, counting multiplicity. This favorable situation

can be duplicated for any ļ¬eld F , that is, we can construct an algebraic extension C of F

with the property that any polynomial in C[X] splits over C. There are many ways to

express this idea.

3.3.1 Proposition

If C is a ļ¬eld, the following conditions are equivalent:

(1) Every nonconstant polynomial f ā C[X] has at least one root in C.

(2) Every nonconstant polynomial f ā C[X] splits over C.

(3) Every irreducible polynomial f ā C[X] is linear.

(4) C has no proper algebraic extensions.

If any (and hence all) of these conditions are satisļ¬ed, we say that C is algebraically closed.

Proof. (1) implies (2): By (1) we may write f = (X ā’ Ī±1 )g. Proceed inductively to show

that any nonconstant polynomial is a product of linear factors.

(2) implies (3): If f is an irreducible polynomial in C[X], then by (2.9.1), f is non-

constant. By (2), f is a product of linear factors. But f is irreducible, so there can be

only one such factor.

(3) implies (4): Let E be an algebraic extension of C. If Ī± ā E, let f be the minimal

polynomial of Ī± over C. Then f is irreducible and by (3), f is of the form X ā’ Ī±. But

then Ī± ā C, so E = C.

3.3. ALGEBRAIC CLOSURES 9

(4) implies (1): Let f be a nonconstant polynomial in C[X], and adjoin a root Ī± of f

to obtain C(Ī±), as in (3.1.3). But then C(Ī±) is an algebraic extension of C, so by (4),

Ī± ā C. ā™£

It will be useful to embed an arbitrary ļ¬eld F in an algebraically closed ļ¬eld.

3.3.2 Deļ¬nitions and Comments

An extension C of F is an algebraic closure of F if C is algebraic over F and C is

algebraically closed.

Note that C is minimal among algebraically closed extensions of F . For if F ā¤ K ā¤ C

and Ī± ā C, Ī± ā K, then since Ī± is algebraic over F it is algebraic over K. But since

/

Ī± ā K, the minimal polynomial of Ī± over K is a nonlinear irreducible polynomial in

/

K[X]. By (3) of (3.3.1), K cannot be algebraically closed.

If C is an algebraic extension of F , then in order for C to be an algebraic closure of F

it is suļ¬cient that every polynomial in F [X] (rather than C[X]) splits over C. To prove

this, we will need the following result.

3.3.3 Proposition

If E is generated over F by ļ¬nitely many elements Ī±1 , . . . , Ī±n algebraic over F (so that

E = F (Ī±1 , . . . , Ī±n )), then E is a ļ¬nite extension of F .

Proof. Set E0 = F and Ek = F (Ī±1 , . . . , Ī±k ), 1 ā¤ k ā¤ n (so En = E). Then Ek =

Ekā’1 (Ī±k ), where Ī±k is algebraic over F and hence over Ekā’1 . But by (3.1.7), [Ek : Ekā’1 ]

is the degree of the minimal polynomial of Ī±k over Ekā’1 , which is ļ¬nite. By (3.1.9),

n

[E : F ] = k=1 [Ek : Ekā’1 ] < ā. ā™£

3.3.4 Corollary

If E is an extension of F and A is the set of all elements in E that are algebraic over F

(the algebraic closure of F in E), then A is a subļ¬eld of E.

Proof. If Ī±, Ī² ā A, then the sum, diļ¬erence, product and quotient (if Ī² = 0) of Ī± and Ī²

belong to F (Ī±, Ī²), which is a ļ¬nite extension of F by (3.3.3), and therefore an algebraic

extension of F by (3.1.10). But then Ī± + Ī², Ī± ā’ Ī², Ī±Ī² and Ī±/Ī² belong to A, proving that

A is a ļ¬eld. ā™£

3.3.5 Corollary (Transitivity of Algebraic Extensions)

If E is algebraic over K (in other words, every element of E is algebraic over K), and K

is algebraic over F , then E is algebraic over F .

Proof. Let Ī± ā E, and let m(X) = b0 + b1 X + Ā· Ā· Ā· + bnā’1 X nā’1 + X n be the minimal

polynomial of Ī± over K. The bi belong to K and are therefore algebraic over F . If

L = F (b0 , b1 , . . . , bnā’1 ), then by (3.3.3), L is a ļ¬nite extension of F . Since the coeļ¬cients

of m(X) belong to L, Ī± is algebraic over L, so by (3.1.7), L(Ī±) is a ļ¬nite extension of L.

By (3.1.9), L(Ī±) is a ļ¬nite extension of F . By (3.1.10), Ī± is algebraic over F . ā™£

10 CHAPTER 3. FIELD FUNDAMENTALS

Now we can add another condition to (3.3.1).

3.3.6 Proposition

Let C be an algebraic extension of F . Then C is an algebraic closure of F if and only if

every nonconstant polynomial in F [X] splits over C.

Proof. The āonly ifā part follows from (2) of (3.3.1), since F ā C. Thus assume that

every nonconstant polynomial in F [X] splits over C. If f is a nonconstant polynomial

in C[X], we will show that f has at least one root in C, and it will follow from (1) of

(3.3.1) that C is algebraically closed. Adjoin a root Ī± of f to obtain the extension C(Ī±).

Then C(Ī±) is algebraic over C by (3.1.7), and C is algebraic over F by hypothesis. By

(3.3.5), C(Ī±) is algebraic over F , so Ī± is algebraic over F . But then Ī± is a root of some

polynomial g ā F [X], and by hypothesis, g splits over C. By deļ¬nition of āsplitsā (see

(3.2.1)), all roots of g lie in C, in particular Ī± ā C. Thus f has at least one root in C. ā™£

To avoid a lengthy excursion into formal set theory, we argue intuitively to establish

the following three results. (For complete proofs, see the appendix to Chapter 3.)

3.3.7 Theorem

Every ļ¬eld F has an algebraic closure.

Informal argument. Well-order F [X] and use transļ¬nite induction, beginning with the

ļ¬eld F0 = F . At stage f we adjoin all roots of the polynomial f by constructing a

splitting ļ¬eld for f over the ļ¬eld F<f that has been generated so far by the recursive

procedure. When we reach the end of the process, we will have a ļ¬eld C such that every

polynomial f in F [X] splits over C. By (3.3.6), C is an algebraic closure of F . ā™£

3.3.8 Theorem

Any two algebraic closures C and C of F are F -isomorphic.

Informal argument. Carry out the recursive procedure described in (3.3.7) in both C

and C . At each stage we may use the fact that any two splitting ļ¬elds of the same

polynomial are F -isomorphic; see (3.2.5). When we ļ¬nish, we have F -isomorphic algebraic

closures of F , say D ā C and D ā C . But an algebraic closure is a minimal algebraically

closed extension by (3.3.2), and therefore D = C and D = C . ā™£

3.3.9 Theorem

If E is an algebraic extension of F , C is an algebraic closure of F , and i is an embedding

(that is, a monomorphism) of F into C, then i can be extended to an embedding of E

into C.

3.4. SEPARABILITY 11

Informal argument. Each Ī± ā E is a root of some polynomial in F [X], so if we allow Ī± to

range over all of E, we get a collection S of polynomials in F [X]. Within C, carry out the

recursive procedure of (3.3.7) on the polynomials in S. The resulting ļ¬eld lies inside C

and contains an F -isomorphic copy of E. ā™£

Problems For Section 3.3

1. Show that the converse of (3.3.3) holds, that is, if E is a ļ¬nite extension of F , then E

is generated over F by ļ¬nitely many elements that are algebraic over F .

2. An algebraic number is a complex number that is algebraic over the rational ļ¬eld Q.

A transcendental number is a complex number that is not algebraic over Q. Show that

there only countably many algebraic numbers, and consequently there are uncountably

many transcendental numbers.

3. Give an example of an extension C/F such that C is algebraically closed but C is not

an algebraic extension of F .

4. Give an example of an extension E/F such that E is an algebraic but not a ļ¬nite

extension of F .

5. In the proof of (3.3.7), why is C algebraic over F ?

6. Show that the set A of algebraic numbers is an algebraic closure of Q.

7. If E is an algebraic extension of the inļ¬nite ļ¬eld F , show that |E| = |F |.

8. Show that any set S of nonconstant polynomials in F [X] has a splitting ļ¬eld over F .

9. Show that an algebraically closed ļ¬eld must be inļ¬nite.

3.4 Separability

If f is a polynomial in F [X], we can construct a splitting ļ¬eld K for f over F , and all

roots of f must lie in K. In this section we investigate the multiplicity of the roots.

3.4.1 Deļ¬nitions and Comments

An irreducible polynomial f ā F [X] is separable if f has no repeated roots in a splitting

ļ¬eld; otherwise f is inseparable. If f is an arbitrary polynomial, not necessarily irreducible,

then we call f separable if each of its irreducible factors is separable.

Thus if f (X) = (X ā’ 1)2 (X ā’ 3) over Q, then f is separable, because the irreducible

factors (X ā’ 1) and (X ā’ 3) do not have repeated roots. We will see shortly that over a

ļ¬eld of characteristic 0 (for example, the rationals), every polynomial is separable. Here

is a method for testing for multiple roots.

3.4.2 Proposition

If

f (X) = a0 + a1 X + Ā· Ā· Ā· + an X n ā F [X],

12 CHAPTER 3. FIELD FUNDAMENTALS

let f be the derivative of f , deļ¬ned by

f (X) = a1 + 2a2 X + Ā· Ā· Ā· + nan X nā’1 .

[Note that the derivative is a purely formal expression; we completely ignore questions

about existence of limits. One can check by brute force that the usual rules for diļ¬eren-

tiating a sum and product apply].

If g is the greatest common divisor of f and f , then f has a repeated root in a splitting

ļ¬eld if and only if the degree of g is at least 1.

Proof. If f has a repeated root, we can write f (X) = (X ā’Ī±)r h(X) where r ā„ 2. Applying

the product rule for derivatives, we see that (X ā’ Ī±) is a factor of both f and f , and

consequently deg g ā„ 1. Conversely, if deg g ā„ 1, let Ī± be a root of g in some splitting

ļ¬eld. Then (X ā’ Ī±) is a factor of both f and f . We will show that Ī± is a repeated root

of f . If not, we may write f (X) = (X ā’ Ī±)h(X) where h(Ī±) = 0. Diļ¬erentiate to obtain

f (X) = (X ā’ Ī±)h (X) + h(X), hence f (Ī±) = h(Ī±) = 0. This contradicts the fact that

(X ā’ Ī±) is a factor of f . ā™£

3.4.3 Corollary

(1) Over a ļ¬eld of characteristic zero, every polynomial is separable.

(2) Over a ļ¬eld F of prime characteristic p, the irreducible polynomial f is inseparable

if and only if f is the zero polynomial. Equivalently, f is a polynomial in X p ; we

abbreviate this as f ā F [X p ].

Proof. (1) Without loss of generality, we can assume that we are testing an irreducible

polynomial f . The derivative of X n is nX nā’1 , and in a ļ¬eld of characteristic 0, n cannot

be 0. Thus f is a nonzero polynomial whose degree is less than that of f . Since f is

irreducible, the gcd of f and f is either 1 or f , and the latter is excluded because f

cannot possibly divide f . By (3.4.2), f is separable.

(2) If f = 0, the argument of (1) shows that f is separable. If f = 0, then

gcd(f, f ) = f , so by (3.4.2), f is inseparable. In characteristic p, an integer n is zero if

and only if n is a multiple of p, and it follows that f = 0 iļ¬ f ā F [X p ]. ā™£

By (3.4.3), part (1), every polynomial over the rationals (or the reals or the complex

numbers) is separable. This pleasant property is shared by ļ¬nite ļ¬elds as well. First

note that a ļ¬nite ļ¬eld F cannot have characteristic 0, since a ļ¬eld of characteristic 0

must contain a copy of the integers (and the rationals as well), and we cannot squeeze

inļ¬nitely many integers into a ļ¬nite set. Now recall the binomial expansion modulo p,

p

which is simply (a + b)p = ap + bp , since p divides ( k ) for 1 ā¤ k ā¤ p ā’ 1. [By induction,

n n n

(a + b)p = ap + bp for every positive integer n.] Here is the key step in the analysis.

3.4.4 The Frobenius Automorphism

Let F be a ļ¬nite ļ¬eld of characteristic p, and deļ¬ne f : F ā’ F by f (Ī±) = Ī±p . Then f is

an automorphism. In particular, if Ī± ā F then Ī± = Ī² p for some Ī² ā F .

3.4. SEPARABILITY 13

Proof. We have f (1) = 1 and

f (Ī± + Ī²) = (Ī± + Ī²)p = Ī±p + Ī² p = f (Ī±) + f (Ī²),

f (Ī±Ī²) = (Ī±Ī²)p = Ī±p Ī² p = f (Ī±)f (Ī²)

so f is a monomorphism. But an injective function from a ļ¬nite set to itself is automati-

cally surjective, and the result follows. ā™£

3.4.5 Proposition

Over a ļ¬nite ļ¬eld, every polynomial is separable.

Proof. Suppose that f is an irreducible polynomial over the ļ¬nite ļ¬eld F with repeated

roots in a splitting ļ¬eld. By (3.4.3), part (2), f (X) has the form a0 + a1 X p + Ā· Ā· Ā· + an X np

with the ai ā F . By (3.4.4), for each i there is an element bi ā F such that bp = ai . But

i

then

(b0 + b1 X + Ā· Ā· Ā· + bn X n )p = bp + bp X p + Ā· Ā· Ā· + bp X np = f (X)

n

0 1

which contradicts the irreducibility of f . ā™£

Separability of an element can be deļ¬ned in terms of its minimal polynomial.

3.4.6 Deļ¬nitions and Comments

If E is an extension of F and Ī± ā E, then Ī± is separable over F if Ī± is algebraic over F

and min(Ī±, F ) is a separable polynomial. If every element of E is separable over F , we say

that E is a separable extension of F or the extension E/F is separable or E is separable

over F . By (3.4.3) and (3.4.5), every algebraic extension of a ļ¬eld of characteristic zero

or a ļ¬nite ļ¬eld is separable.

3.4.7 Lemma

If F ā¤ K ā¤ E and E is separable over F , then K is separable over F and E is separable

over K.

Proof. Since K is a subļ¬eld of E, K/F is separable. If Ī± ā E, then since Ī± is a root of

min(Ī±, F ), it follows from (1) of (3.1.6) that min(Ī±, K) divides min(Ī±, F ). By hypothesis,

min(Ī±, F ) has no repeated roots in a splitting ļ¬eld, so neither does min(Ī±, K). Thus E/K

is separable. ā™£

The converse of (3.4.7) is also true: If K/F and E/K are separable, then E/F is

separable. Thus we have transitivity of separable extensions. We will prove this (for ļ¬nite

extensions) in the exercises.

In view of (3.4.6), we can produce many examples of separable extensions. Inseparable

extensions are less common, but here is one way to construct them.

14 CHAPTER 3. FIELD FUNDAMENTALS

3.4.8 Example

Let F = Fp (t) be the set of rational functions (in the indeterminate t) with coeļ¬cients

in the ļ¬eld with p elements (the integers mod p). Thus an element of F looks like

a0 + a1 t + Ā· Ā· Ā· + am tm

.

b0 + b1 t + Ā· Ā· Ā· + b n t n

ā

with the ai and bj in Fp . Adjoin p t, that is, a root of X p ā’ t, to create the extension E.

Note that X p ā’ t is irreducible by Eisenstein, because t is irreducible in Fp [t]. (The

product of two nonconstant polynomials in t cannot possibly be t.) The extension E/F

is inseparable, since

ā ā

X p ā’ t = X p ā’ ( t)p = (X ā’ t)p ,

p p

which has multiple roots.

Problems For Section 3.4

1. Give an example of a separable polynomial f whose derivative is zero. (In view of

(3.4.3), f cannot be irreducible.)

2. Let Ī± ā E, where E is an algebraic extension of a ļ¬eld F of prime characteristic p.

Let m(X) be the minimal polynomial of Ī± over the ļ¬eld F (Ī±p ). Show that m(X)

splits over E, and in fact Ī± is the only root, so that m(X) is a power of (X ā’ Ī±).

3. Continuing Problem 2, if Ī± is separable over the ļ¬eld F (Ī±p ), show that Ī± ā F (Ī±p ).

4. A ļ¬eld F is said to be perfect if every polynomial over F is separable. Equivalently,

every algebraic extension of F is separable. Thus ļ¬elds of characteristic zero and

ļ¬nite ļ¬elds are perfect. Show that if F has prime characteristic p, then F is perfect

if and only if every element of F is the pth power of some element of F . For short we

write F = F p .

In Problems 5-8, we turn to transitivity of separable extensions.

5. Let E be a ļ¬nite extension of a ļ¬eld F of prime characteristic p, and let K = F (E p )

be the subļ¬eld of E obtained from F by adjoining the pth powers of all elements of

E. Show that F (E p ) consists of all ļ¬nite linear combinations of elements in E p with

coeļ¬cients in F .

6. Let E be a ļ¬nite extension of the ļ¬eld F of prime characteristic p, and assume that

E = F (E p ). If the elements y1 , . . . , yr ā E are linearly independent over F , show

p p

that y1 , . . . , yr are linearly independent over F .

7. Let E be a ļ¬nite extension of the ļ¬eld F of prime characteristic p. Show that the

extension is separable if and only if E = F (E p ).

8. If F ā¤ K ā¤ E with [E : F ] < ā, with E separable over K and K separable over F ,

show that E is separable over F .

9. Let f be an irreducible polynomial in F [X], where F has characteristic p > 0. Express

m

f (X) as g(X p ), where the nonnegative integer m is a large as possible. (This makes

0

sense because X p = X, so m = 0 always works, and f has ļ¬nite degree, so m is

bounded above.) Show that g is irreducible and separable.

3.5. NORMAL EXTENSIONS 15

m

10. Continuing Problem 9, if f has only one distinct root Ī±, show that Ī±p ā F .

11. If E/F , where char F = p > 0, and the element Ī± ā E is algebraic over F , show that

n

the minimal polynomial of Ī± over F has only one distinct root if and only if Ī±p ā F

for some nonnegative integer n. (In this case we say that Ī± is purely inseparable over

F .)

3.5 Normal Extensions

Let E/F be a ļ¬eld extension. In preparation for Galois theory, we are going to look at

monomorphisms deļ¬ned on E, especially those which ļ¬x F . First we examine what an

F -monomorphism does to the roots of a polynomial in F [X].

3.5.1 Lemma

Let Ļ : E ā’ E be an F -monomorphism, and assume that the polynomial f ā F [X] splits

over E. If Ī± is a root of f in E, then so is Ļ(Ī±). Thus Ļ permutes the roots of f .

Proof. If b0 + b1 Ī± + Ā· Ā· Ā· + bn Ī±n = 0, with the bi ā F , apply Ļ and note that since Ļ is an

F -monomorphism, Ļ(bi ) = bi and Ļ(Ī±i ) = (Ļ(Ī±))i . Thus

b0 + b1 Ļ(Ī±) + Ā· Ā· Ā· + bn (Ļ(Ī±))n = 0. ā™£

Now let C be an algebraic closure of E. It is convenient to have C available because

it will contain all the roots of a polynomial f ā E[X], even if f does not split over E. We

are going to count the number of embeddings of E in C that ļ¬x F , that is, the number

of F -monomorphisms of E into C. Here is the key result.

3.5.2 Theorem

Let E/F be a ļ¬nite separable extension of degree n, and let Ļ be an embedding of F

in C. Then Ļ extends to exactly n embeddings of E in C; in other words, there are

exactly n embeddings Ļ„ of E in C such that the restriction Ļ„ |F of Ļ„ to F coincides

with Ļ. In particular, taking Ļ to be the identity function on F , there are exactly n

F -monomorphisms of E into C.

Proof. An induction argument works well. If n = 1 then E = F and there is nothing to

prove, so assume n > 1 and choose an element Ī± that belongs to E but not to F . If f is

the minimal polynomial of Ī± over F , let g = Ļ(f ). (This is a useful shorthand notation,

indicating that if ai is one of the coeļ¬cients of f , the corresponding coeļ¬cient of g is

Ļ(ai ).) Any factorization of g can be translated via the inverse of Ļ to a factorization

of f , so g is separable and irreducible over the ļ¬eld Ļ(F ). If Ī² is any root of g, then there

is a unique isomorphism of F (Ī±) and (Ļ(F ))(Ī²) that carries Ī± into Ī² and coincides with

Ļ on F . Explicitly,

b0 + b1 Ī± + Ā· Ā· Ā· + br Ī±r ā’ Ļ(b0 ) + Ļ(b1 )Ī² + Ā· Ā· Ā· + Ļ(br )Ī² r .

16 CHAPTER 3. FIELD FUNDAMENTALS

Now if deg g = r, then [F (Ī±) : F ] = deg f = deg g = r as well, so by (3.1.9), [E : F (Ī±)] =

n/r < n. By separability, g has exactly r distinct roots in C, so there are exactly r

possible choices of Ī². In each case, by the induction hypothesis,the resulting embedding

of F (Ī±) in C has exactly n/r extensions to embeddings of E in C. This produces n

distinct embeddings of E in C extending Ļ. But if Ļ„ is any embedding of F in C that

extends Ļ, then just as in (3.5.1), Ļ„ must take Ī± to a root of g, i.e., to one of the Ī²ā™s.

If there were more than n possible Ļ„ ā™s, there would have to be more than n/r possible

extensions of at least one of the embeddings of F (Ī±) in C. This would contradict the

induction hypothesis. ā™£

3.5.3 Example

ā

Q(

Adjoin the positive cube root of 2 to the rationals to get E = ā 3 2). The roots of the

ā ā

irreducible polynomial f (X) = X 3 ā’ 2 are 3 2, Ļ 3 2 and Ļ 2 3 2, where Ļ = ei2Ļ/3 =

ā ā

ā’ 1 + 1 i 3 and Ļ 2 = ei4Ļ/3 = ā’ 1 ā’ 1 i 3.

2 2 2 2

Notice that the polynomial f has a root in E but does not split in E (because the

other two roots are complex and E consists entirely of real numbers). We give a special

name to extensions that do not have this annoying drawback.

3.5.4 Deļ¬nition

The algebraic extension E/F is normal (we also say that E is normal over F ) if every

irreducible polynomial over F that has at least one root in E splits over E. In other

words, if Ī± ā E, then all conjugates of Ī± over F (i.e., all roots of the minimal polynomial

of Ī± over F ) belong to E.

Here is an equivalent condition.

3.5.5 Theorem

The ļ¬nite extension E/F is normal if and only if every F -monomorphism of E into an

algebraic closure C is actually an F -automorphism of E. (The hypothesis that E/F is

ļ¬nite rather than simply algebraic can be removed, but we will not need the more general

result.)

Proof. If E/F is normal, then as in (3.5.1), an F -monomorphism Ļ„ of E into C must map

each element of E to one of its conjugates. Thus by hypothesis, Ļ„ (E) ā E. But Ļ„ (E) is

an isomorphic copy of E , so it must have the same degree as E over F . Since the degree

is assumed ļ¬nite, we have Ļ„ (E) = E. (All we are saying here is that an m-dimensional

subspace of an m-dimensional vector space is the entire space.) Conversely, let Ī± ā E, and

let Ī² be any conjugate of Ī± over F . As in the proof of (3.5.2), there is an F -monomorphism

of E into C that carries Ī± to Ī². If all such embeddings are F -automorphisms of E, we

must have Ī² ā E, and we conclude that E is normal over F . ā™£

3.5.6 Remarks

In (3.5.2) and (3.5.5), the algebraic closure can be replaced by any ļ¬xed normal extension

of F containing E; the proof is the same. Also, the implication Ļ„ (E) ā E ā’ Ļ„ (E) = E

3.5. NORMAL EXTENSIONS 17

holds for any F -monomorphism Ļ„ and any ļ¬nite extension E/F ; normality is not involved.

The next result yields many explicit examples of normal extensions.

3.5.7 Theorem

The ļ¬nite extension E/F is normal if and only if E is a splitting ļ¬eld for some polynomial

f ā F [X].

Proof. Assume that E is normal over F . Let Ī±1 , . . . , Ī±n be a basis for E over F , and

let fi be the minimal polynomial of Ī±i over F, i = 1, . . . , n. Since fi has a root Ī±i in E,

fi splits over E, hence so does f = f1 Ā· Ā· Ā· fn . If f splits over a ļ¬eld K with F ā K ā E,

then each Ī±i belongs to K, and therefore K must coincide with E. Thus E is a splitting

ļ¬eld for f over F . Conversely, let E be a splitting ļ¬eld for f over F , where the roots of f

are Ī±i , i = 1, . . . , n. Let Ļ„ be an F -monomorphism of E into an algebraic closure. As in

(3.5.1), Ļ„ takes each Ī±i into another root of f , and therefore Ļ„ takes a polynomial in the

Ī±i to another polynomial in the Ī±i . But F (Ī±1 , . . . , Ī±n ) = E, so Ļ„ (E) ā E. By (3.5.6), Ļ„

is an automorphism of E, so by (3.5.5), E/F is normal. ā™£

3.5.8 Corollary

Let F ā¤ K ā¤ E, where E is a ļ¬nite extension of F . If E/F is normal, so is E/K.

Proof. By (3.5.7), E is a splitting ļ¬eld for some polynomial f ā F [X], so that E is

generated over F by the roots of f . But then f ā K[X] and E is generated over K by

the roots of f . Again by (3.5.7), E/K is normal. ā™£

3.5.9 Deļ¬nitions and Comments

If E/F is normal and separable, it is said to be a Galois extension; we also say that E is

Galois over F . It follows from (3.5.2) and (3.5.5) that if E/F is a ļ¬nite Galois extension,

then there are exactly [E : F ] F -automorphisms of E. If E/F is ļ¬nite and separable but

not normal, then at least one F -embedding of E into an algebraic closure must fail to be

an automorphism of E. Thus in this case, the number of F -automorphisms of E is less

than the degree of the extension.

If E/F is an arbitrary extension, the Galois group of the extension, denoted by

Gal(E/F ), is the set of F -automorphisms of E. (The set is a group under composition of

functions.)

3.5.10 Example

ā

Let E = Q( 3 2), as in (3.5.3). The Galois group of the extension consists of the identity

ā

automorphism alone. For any Q-monomorphism Ļ of E must take 3 ā into a root of

2

X ā’ 2. Since the other two roots are complex and do not belong to E, 3 2 must map to

3

ā

itself. But Ļ is completely determined by its action on 3 2, and the result follows.

If E/F is not normal, we can always enlarge E to produce a normal extension of F .

If C is an algebraic closure of E, then C contains all the roots of every polynomial in

F [X], so C/F is normal. Let us try to look for a smaller normal extension.

18 CHAPTER 3. FIELD FUNDAMENTALS

3.5.11 The Normal Closure

Let E be a ļ¬nite extension of F , say E = F (Ī±1 , . . . , Ī±n ). If N ā E is any normal extension

of F , then N must contain the Ī±i along with all conjugates of the Ī±i , that is, all roots of

min(Ī±i , F ), i = 1, . . . , n. Thus if f is the product of these minimal polynomials, then N

must contain the splitting ļ¬eld K for f over F . But K/F is normal by (3.5.7), so K must

be the smallest normal extension of F that contains E. It is called the normal closure of

E over F .

We close the section with an important result on the structure of ļ¬nite separable

extensions.

3.5.12 Theorem of the Primitive Element

If E/F is a ļ¬nite separable extension, then E = F (Ī±) for some Ī± ā E. We say that Ī± is

a primitive element of E over F .

Proof. We will argue by induction on n = [E : F ]. If n = 1 then E = F and we can take

Ī± to be any member of F . If n > 1, choose Ī± ā E \ F . By the induction hypothesis, there

is a primitive element Ī² for E over F (Ī±), so that E = F (Ī±, Ī²). We are going to show

that if c ā F is properly chosen, then E = F (Ī± + cĪ²). Now by (3.5.2), there are exactly n

F -monomorphisms of E into an algebraic closure C, and each of these maps restricts to

an F -monomorphism of F (Ī± + cĪ²) into C. If F (Ī± + cĪ²) = E, then [F (Ī± + cĪ²) : F ] < n,

and it follows from (3.5.2) that at least two embeddings of E, say Ļ and Ļ„ , must coincide

when restricted. Therefore

Ļ(Ī±) + cĻ(Ī²) = Ļ„ (Ī±) + cĻ„ (Ī²),

hence

Ļ(Ī±) ā’ Ļ„ (Ī±)

c= . (1)

Ļ„ (Ī²) ā’ Ļ(Ī²)

(If Ļ„ (Ī²) = Ļ(Ī²) then by the previous equation, Ļ„ (Ī±) = Ļ(Ī±). But an F -embedding

of E is determined by what it does to Ī± and Ī², hence Ļ = Ļ„ , a contradiction.) Now

an F -monomorphism must map Ī± to one of its conjugates over F , and similarly for Ī².

Thus there are only ļ¬nitely many possible values for the ratio in (1). If we select c to

be diļ¬erent from each of these values, we reach a contradiction of our assumption that

F (Ī± + cĪ²) = E. The proof is complete if F is an inļ¬nite ļ¬eld. We must leave a gap here,

to be ļ¬lled later (see (6.4.4)). If F is ļ¬nite, then so is E (since E is a ļ¬nite-dimensional

vector space over F ). We will show that the multiplicative group of nonzero elements of

a ļ¬nite ļ¬eld E is cyclic, so if Ī± is a generator of this group, then E = F (Ī±). ā™£

Problems For Section 3.5

1. Give an example of ļ¬elds F ā¤ K ā¤ E such that E/F is normal but K/F is not.

ā

2. Let E = Q( a), where a is an integer that is not a perfect square. Show that E/Q is

normal.

3.5. NORMAL EXTENSIONS 19

3. Give an example of ļ¬elds F ā¤ K ā¤ E such that E/K and K/F are normal, but E/F

is not. Thus transitivity fails for normal extensions.

4. Suppose that in (3.5.2), the hypothesis of separability is dropped. State and prove an

appropriate conclusion.

āā

5. Show that E = Q( 2, 3) is a Galois extension of Q.

6. In Problem 5, ļ¬nd the Galois group of E/Q.

7. Let E be a ļ¬nite extension of F , and let K be a normal closure (= minimal normal

extension) of E over F , as in (3.5.11). Is K unique?

8. If E1 and E2 are normal extensions of F , show that E1 ā© E2 is normal over F .

Appendix To Chapter 3

In this appendix, we give a precise development of the results on algebraic closure treated

informally in the text.

A3.1 Lemma

Let E be an algebraic extension of F , and let Ļ : E ā’ E be an F -monomorphism. Then

Ļ is surjective, hence Ļ is an automorphism of E.

Proof. Let Ī± ā E, and let f (X) be the minimal polynomial of Ī± over F . We consider the

subļ¬eld L of E generated over F by the roots of f that lie in E. Then L is an extension

of F that is ļ¬nitely generated by algebraic elements, so by (3.3.3), L/F is ļ¬nite. As in

(3.5.1), Ļ takes a root of f to a root of f , so Ļ(L) ā L. But [L : F ] = [Ļ(L) : F ] < ā

(Ļ maps a basis to a basis), and consequently Ļ(L) = L. But Ī± ā L, so Ī± ā Ļ(L). ā™£

The following result, due to Artin, is crucial.

A3.2 Theorem

If F is any ļ¬eld, there is an algebraically closed ļ¬eld E containing F .

Proof. For each nonconstant polynomial f in F [X], we create a variable X(f ). If T is the

collection of all such variables, we can form the ring F [T ] of all polynomials in all possible

ļ¬nite sets of variables in T , with coeļ¬cients in F . Let I be the ideal of F [T ] generated

by the polynomials f (X(f )), f ā F [X]. We claim that I is a proper ideal. If not,

then 1 ā I, so there are ļ¬nitely many polynomials f1 , . . . , fn in F [X] and polynomials

n

h1 , . . . , hn in F [T ] such that i=1 hi fi (X(fi )) = 1. Now only ļ¬nitely many variables

Xi = X(fi ), i = 1, . . . , m, can possibly appear in the hi , so we have an equation of the

form

n

hi (X1 , . . . , Xm )fi (Xi ) = 1 (1)

i=1

20 CHAPTER 3. FIELD FUNDAMENTALS

where m ā„ n. Let L be the extension of F formed by successively adjoining the roots of

f1 , . . . , fn . Then each fi has a root Ī±i ā L. If we set Ī±i = 0 for n ā¤ i < m and then set

Xi = Ī±i for each i in (1), we get 0 = 1, a contradiction.

Thus the ideal I is proper, and is therefore contained in a maximal ideal M. Let E1

be the ļ¬eld F [T ]/M. Then E1 contains an isomorphic copy of F , via the map taking

a ā F to a + M ā E1 . (Note that if a ā M, a = 0, then 1 = aā’1 a ā M, a contradiction.)

Consequently, we can assume that F ā¤ E1 . If f is any nonconstant polynomial in F [X],

then X(f ) + M ā E1 and f (X(f ) + M) = f (X(f )) + M = 0 because f (X(f )) ā I ā M.

Iterating the above procedure, we construct a chain of ļ¬elds F ā¤ E1 ā¤ E2 ā¤ Ā· Ā· Ā· such

that every polynomial of degree at least 1 in En [X] has a root in En+1 . The union E of

all the En is a ļ¬eld, and every nonconstant polynomial f in E[X] has all its coeļ¬cients

in some En . Therefore f has a root in En+1 ā E. ā™£

A3.3 Theorem

Every ļ¬eld F has an algebraic closure.

Proof. By (3.5.12), F has an algebraically closed extension L. If E is the algebraic closure

of F in L (see 3.3.4), then E/F is algebraic. Let f be a nonconstant polynomial in E[X].

Then f has a root Ī± in L (because L is algebraically closed). We now have Ī± algebraic

over E (because f ā E[X]), and E algebraic over F . As in (3.3.5), Ī± is algebraic over F ,

hence Ī± ā E. By (3.3.1), E is algebraically closed. ā™£

A3.4 Problem

Suppose that Ļ is a monomorphism of F into the algebraically closed ļ¬eld L. Let E be

an algebraic extension of F , and Ī± an element of E with minimal polynomial f over F .

We wish to extend Ļ to a monomorphism from F (Ī±) to L. In how many ways can this

be done?

Let Ļf be the polynomial in (ĻF )[X] obtained from f by applying Ļ to the coeļ¬cients

of f . Any extension of f is determined by what it does to Ī±, and as in (3.5.1), the image

of Ī± is a root of Ļf . Now the number of distinct roots of f in an algebraic closure of F ,

call it t, is the same as the number of distinct roots of Ļf in L; this follows from the

isomorphism extension theorem (3.2.5). Thus the number of extensions is at most t.

But if Ī² is any root of Ļf , we can construct an extension of Ļ by mapping the element

h(Ī±) ā F (Ī±) to (Ļh)(Ī²); in particular, Ī± is mapped to Ī². To show that the deļ¬nition

makes sense, suppose that h1 (Ī±) = h2 (Ī±). Then (h1 ā’ h2 )(Ī±) = 0, so f divides h1 ā’ h2

in F [X]. Consequently, Ļf divides Ļh1 ā’ Ļh2 in (ĻF )[X], so (Ļh1 )(Ī²) = (Ļh2 )(Ī²).

We conclude that the number of extensions of Ļ is the number of distinct roots of f

in an algebraic closure of F .

Rather than extend Ļ one element at a time, we now attempt an extension to all of E.

A3.5 Theorem

Let Ļ : F ā’ L be a monomorphism, with L algebraically closed. If E is an algebraic

extension of F , then Ļ has an extension to a monomorphism Ļ„ : E ā’ L.

3.5. NORMAL EXTENSIONS 21

Proof. Let G be the collection of all pairs (K, Āµ) where K is an intermediate ļ¬eld between

F and E and Āµ is an extension of Ļ to a monomorphism from K to L. We partially

order G by (K1 , Āµ) ā¤ (K2 , Ļ) iļ¬ K1 ā K2 and Ļ restricted to K1 coincides with Āµ. Since

(F, Ļ) ā G, we have G = ā…. If the pairs (Ki , Āµi ), i ā I, form a chain, there is an upper

bound (K, Āµ) for the chain, where K is the union of the Ki and Āµ coincides with Āµi on

each Ki . By Zornā™s lemma, G has a maximal element (K0 , Ļ„ ). If K0 ā‚ E, let Ī± ā E \ K0 .

By (3.5.12), Ļ„ has an extension to K0 (Ī±), contradicting maximality of (K0 , Ļ„ ). ā™£

A3.6 Corollary

In (3.5.12), if E is algebraically closed and L is algebraic over Ļ(F ), then Ļ„ is an isomor-

phism.

Proof. Since E is algebraically closed, so is Ļ„ (E). Since L is algebraic over Ļ(F ), it is

algebraic over the larger ļ¬eld Ļ„ (E). By (1) āā’ (4) in (3.3.1), L = Ļ„ (E). ā™£

A3.7 Theorem

Any two algebraic closures L and E of a ļ¬eld F are F -isomorphic.

Proof. We can assume that F is a subļ¬eld of L and Ļ : F ā’ L is the inclusion map.

By (3.5.12), Ļ extends to an isomorphism Ļ„ of E and L, and since Ļ„ is an extension of Ļ,

it is an F -monomorphism. ā™£

A3.8 Theorem (=Theorem 3.3.9)

If E is an algebraic extension of F and C is an algebraic closure of F , then any embedding

of F into C can be extended to an embedding of E into C.

Proof. Repeat the proof of (3.5.12), with the mapping Āµ required to be an embedding. ā™£

A3.9 Remark

The argument just given assumes that E is a subļ¬eld of C. This can be assumed without

loss of generality, by (3.5.12), (3.5.12) and (3.5.12). In other words, we can assume that

an algebraic closure of F contains a speciļ¬ed algebraic extension of F .

A3.10 Theorem

Let E be an algebraic extension of F , and let L be the algebraic closure of F containing E

(see 3.5.12). If Ļ is an F -monomorphism from E to L, then Ļ can be extended to an

automorphism of L.

Proof. We have L algebraically closed and L/E algebraic, so by (3.5.12) with E replaced

by L and F by E, Ļ extends to a monomorphism from L to L, an F -monomorphism by

hypothesis. The result follows from (3.5.12). ā™£

Chapter 4

Module Fundamentals

4.1 Modules and Algebras

4.1.1 Deļ¬nitions and Comments

A vector space M over a ļ¬eld R is a set of objects called vectors, which can be added,

subtracted and multiplied by scalars (members of the underlying ļ¬eld). Thus M is an

abelian group under addition, and for each r ā R and x ā M we have an element rx ā M .

Scalar multiplication is distributive and associative, and the multiplicative identity of the

ļ¬eld acts as an identity on vectors. Formally,

r(x + y) = rx + ry; (r + s)x = rx + sx; r(sx) = (rs)x; 1x = x

for all x, y ā M and r, s ā R. A module is just a vector space over a ring. The formal

deļ¬nition is exactly as above, but we relax the requirement that R be a ļ¬eld, and instead

allow an arbitrary ring. We have written the product rx with the scalar r on the left, and

technically we get a left R-module over the ring R. The axioms of a right R-module are

(x + y)r = xr + yr; x(r + s) = xr + xs; (xs)r = x(sr), x1 = x.

āModuleā will always mean left module unless stated otherwise. Most of the time, there

is no reason to switch the scalars from one side to the other (especially if the underlying

ring is commutative). But there are cases where we must be very careful to distinguish

between left and right modules (see Example 6 of (4.1.3)).

4.1.2 Some Basic Properties of Modules

Let M be an R-module. The technique given for rings in (2.1.1) can be applied to establish

the following results, which hold for any x ā M and r ā R. We distinguish the zero vector

0M from the zero scalar 0R .

(1) r0M = 0M [r0M = r(0M + 0M ) = r0M + r0M ]

(2) 0R x = 0M [0R x = (0R + 0R )x = 0R x + 0R x]

1

2 CHAPTER 4. MODULE FUNDAMENTALS

(3) (ā’r)x = r(ā’x) = ā’(rx) [as in (2) of (2.1.1) with a replaced by r and b by x]

(4) If R is a ļ¬eld, or more generally a division ring, then rx = 0M implies that either

r = 0R or x = 0M . [If r = 0, multiply the equation rx = 0M by rā’1 .]

4.1.3 Examples

1. If M is a vector space over the ļ¬eld R, then M is an R-module.

2. Any ring R is a module over itself. Rather than check all the formal requirements,

think intuitively: Elements of a ring can be added and subtracted, and we can

certainly multiply r ā R by x ā R, and the usual rules of arithmetic apply.

3. If R is any ring, then Rn , the set of all n-tuples with components in R, is an

R-module, with the usual deļ¬nitions of addition and scalar multiplication (as in

Euclidean space, e.g., r(x1 , . . . , xn ) = (rx1 , . . . , rxn ), etc).

4. Let M = Mmn (R) be the set of all m Ć— n matrices with entries in R. Then M is

an R-module, where addition is ordinary matrix addition, and multiplication of the

scalar c by the matrix A means multiplication of each entry of A by c.

5. Every abelian group A is a Z-module. Addition and subtraction is carried out

according to the group structure of A; the key point is that we can multiply x ā A

by the integer n. If n > 0, then nx = x + x + Ā· Ā· Ā· + x (n times); if n < 0, then

nx = ā’x ā’ x ā’ Ā· Ā· Ā· ā’ x (|n| times).

In all of these examples, we can switch from left to right modules by a simple notational

change. This is deļ¬nitely not the case in the next example.

6. Let I be a left ideal of the ring R; then I is a left R-module. (If x ā I and r ā R

then rx (but not necessarily xr) belongs to I.) Similarly, a right ideal is a right

R-module, and a two-sided ideal is both a left and a right R-module.

An R-module M permits addition of vectors and scalar multiplication. If multiplica-

tion of vectors is allowed, we have an R-algebra.

4.1.4 Deļ¬nitions and Comments

Let R be a commutative ring. We say that M is an algebra over R, or that M is an

R-algebra, if M is an R-module that is also a ring (not necessarily commutative), and the

ring and module operations are compatible, i.e.,

r(xy) = (rx)y = x(ry) for all x, y ā M and r ā R.

4.1.5 Examples

1. Every commutative ring R is an algebra over itself (see Example 2 of (4.1.3)).

2. An arbitrary ring R is always a Z-algebra (see Example 5 of (4.1.3)).

4.1. MODULES AND ALGEBRAS 3

3. If R is a commutative ring, then Mn (R), the set of all n Ć— n matrices with entries

in R, is an R-algebra (see Example 4 of (4.1.3)).

4. If R is a commutative ring, then the polynomial ring R[X] is an R-algebra, as

is the ring R[[X]] of formal power series; see Examples 5 and 6 of (2.1.3). The

compatibility condition is satisļ¬ed because an element of R can be regarded as a

polynomial of degree 0.

5. If E/F is a ļ¬eld extension, then E is an algebra over F . This continues to hold if E

is a division ring, and in this case we say that E is a division algebra over F .

To check that a subset S of a vector space is a subspace, we verify that S is closed

under addition of vectors and multiplication of a vector by a scalar. Exactly the same

idea applies to modules and algebras.

4.1.6 Deļ¬nitions and Comments

If N is a nonempty subset of the R-module M , we say that N is a submodule of M

(notation N ā¤ M ) if for every x, y ā N and r, s ā R, we have rx + sy ā N . If M is an

R-algebra, we say that N is a subalgebra if N is a submodule that is also a subring.

For example, if A is an abelian group (= Z-module), the submodules of A are the

subsets closed under addition and multiplication by an integer (which amounts to addition

also). Thus the submodules of A are simply the subgroups. If R is a ring, hence a

module over itself, the submodules are those subsets closed under addition and also under

multiplication by any r ā R, in other words, the left ideals. (If we take R to be a right

R-module, then the submodules are the right ideals.)

We can produce many examples of subspaces of vector spaces by considering kernels

and images of linear transformations. A similar idea applies to modules.

4.1.7 Deļ¬nitions and Comments

Let M and N be R-modules. A module homomorphism (also called an R-homomorphism)

from M to N is a map f : M ā’ N such that

f (rx + sy) = rf (x) + sf (y) for all x, y ā M and r, s ā R.

Equivalently, f (x + y) = f (x) + f (y) and f (rx) = rf (x) for all x, y ā M and r ā R.

The kernel of a homomorphism f is ker f = {x ā M : f (x) = 0}, and the image of f

is {f (x) : x ā M }.

If follows from the deļ¬nitions that the kernel of f is a submodule of M , and the image

of f is a submodule of N .

If M and N are R-algebras, an algebra homomorphism or homomorphism of algebras

from M to N is an R-module homomorphism that is also a ring homomorphism.

4.1.8 Another Way to Describe an Algebra

Assume that A is an algebra over the commutative ring R, and consider the map r ā’ r1

of R into A. The commutativity of R and the compatibility of the ring and module

4 CHAPTER 4. MODULE FUNDAMENTALS

operations imply that the map is a ring homomorphism. To see this, note that if r, s ā R

then

(rs)1 = (sr)1 = s(r1) = s[(r1)1] = (r1)(s1).

Furthermore, if y ā A then

(r1)y = r(1y) = r(y1) = y(r1)

so that r1 belongs to the center of A, i.e., the set of elements that commute with everything

in A.

Conversely, if f is a ring homomorphism from the commutative ring R to the center

of the ring A, we can make A into an R-module via rx = f (r)x. The compatibility

conditions are satisļ¬ed because

r(xy) = f (r)(xy) = (f (r)x)y = (rx)y

and

(f (r)x)y = (xf (r))y = x(f (r)y) = x(ry).

Because of this result, the deļ¬nition of an R-algebra is sometimes given as follows. The

ring A is an algebra over the commutative ring R if there exists a ring homomorphism of

R into the center of A. For us at this stage, such a deļ¬nition would be a severe overdose

of abstraction.

Notational Convention: We will often write the module {0} (and the ideal {0} in

a ring) simply as 0.

Problems For Section 4.1

1. If I is an ideal of the ring R, show how to make the quotient ring R/I into a left

R-module, and also show how to make R/I into a right R-module.

2. Let A be a commutative ring and F a ļ¬eld. Show that A is an algebra over F if and

only if A contains (an isomorphic copy of) F as a subring.

Problems 3, 4 and 5 illustrate that familiar properties of vector spaces need not hold

for modules.

3. Give an example of an R-module M with nonzero elements r ā R and x ā M such

that rx = 0.

4. Let M be the additive group of rational numbers. Show that any two elements of M

are linearly dependent (over the integers Z).

5. Continuing Problem 4, show that M cannot have a basis, that is, a linearly independent

spanning set over Z.

6. Prove the modular law for subgroups of a given group G: With the group operation

written multiplicatively,

A(B ā© C) = (AB) ā© C

4.2. THE ISOMORPHISM THEOREMS FOR MODULES 5

if A ā C. Switching to additive notation, we have, for submodules of a given R-module,

A + (B ā© C) = (A + B) ā© C,

again if A ā C.

7. Let T be a linear transformation on the vector space V over the ļ¬eld F . Show how to

make V into an R-module in a natural way, where R is the polynomial ring F [X].

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