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Exactness at E0 no longer holds because the map E0 в†’ E1 need not be injective.
We will use the notation Pв€— в†’ M for a projective resolution, and M в†’ Eв€— for an
injective resolution.

S4.2 Proposition
Every module M has a free (hence projective) resolution.
9

Proof. By (4.3.6), M is a homomorphic image of a free module F0 . Let K0 be the kernel
of the map from F0 onto M . In turn, there is a homomorphism with kernel K1 from a
free module F1 onto K0 , and we have the following diagram:
G K1 G F1 G K0 G F0 GM G0
0
Composing the maps F1 в†’ K0 and K0 в†’ F0 , we get
G K1 G F1 G F0 GM G0
0
which is exact. But now we can п¬Ѓnd a free module F2 and a homomorphism with kernel K2
mapping F2 onto K1 . The above process can be iterated to produce the desired free
resolution. в™Ј
Specifying a module by generators and relations (see (4.6.6) for abelian groups) in-
volves п¬Ѓnding an appropriate F0 and K0 , as in the п¬Ѓrst step of the above iterative process.
Thus a projective resolution may be regarded as a generalization of a speciп¬Ѓcation by gen-
erators and relations.
Injective resolutions can be handled by dualizing the proof of (S4.2).

S4.3 Proposition
Every module M has an injective resolution.
Proof. By (10.7.4), M can be embedded in an injective module E0 . Let C0 be the cokernel
of M в†’ E0 , and map E0 canonically onto C0 . Embed C0 in an injective module E1 , and
let C1 be the cokernel of the embedding map. We have the following diagram:
GM G E0 G C0 G E1 G C1 G0
0
Composing E0 в†’ C0 and C0 в†’ E1 , we have
GM G E0 G E1 G C1 G0
0
which is exact. Iterate to produce the desired injective resolution. в™Ј

S5. Derived Functors
S5.1 Left Derived Functors
Suppose that F is a right exact functor from modules to modules. (In general, the
domain and codomain of F can be abelian categories, but the example we have in mind is
M вЉ—R .) Given a short exact sequence 0 в†’ A в†’ B в†’ C в†’ 0, we form deleted projective
resolutions PAв€— в†’ A, PBв€— в†’ B, PCв€— в†’ C. It is shown in texts on homological algebra
that it is possible to deп¬Ѓne chain maps to produce a short exact sequence of complexes
as shown below.
GA GB GC G0
0 y y y

G PA в€— G PB в€— G PC в€— G0
0
10

The functor F will preserve exactness in the diagram, except at the top row, where we only
have F A в†’ F B в†’ F C в†’ 0 exact. But remember that we are using deleted resolutions,
so that the п¬Ѓrst row is suppressed. The left derived functors of F are deп¬Ѓned by taking
the homology of the complex F (P ), that is,

(Ln F )(A) = Hn [F (PAв€— )].

The word вЂњleftвЂќ is used because the Ln F are computed using left resolutions. It can
be shown that up to natural equivalence, the derived functors are independent of the
particular projective resolutions chosen. By (S3.2), we have the following long exact
sequence:

G (Ln F )(A) G (Ln F )(B) G (Ln F )(C) G (Lnв€’1 F )(A) G В·В·В·
в€‚ в€‚
В·В·В·

S5.2 Right Derived Functors
Suppose now that F is a left exact functor from modules to modules, e.g., HomR (M, ).
We can dualize the discussion in (S5.1) by reversing the vertical arrows in the commuta-
tive diagram of complexes, and replacing projective resolutions such as PAв€— by injective
resolutions EAв€— . The right derived functors of F are deп¬Ѓned by taking the homology of
F (E). In other words,

(Rn F )(A) = H n [F (EAв€— )]

where the superscript n indicates that we are using right resolutions and the indices are
increasing as we move away from the starting point. By (S3.2), we have the following
long exact sequence:
в€‚G G (Rn F )(B) G (Rn F )(C) G (Rn+1 F )(A) G В·В·В·
в€‚
В·В·В· (Rn F )(A)

S5.3 Lemma
(L0 F )(A) в€ј F (A) в€ј (R0 F )(A)
= =
Proof. This is a good illustration of the advantage of deleted resolutions. If Pв€— в†’ A, we
have the following diagram:

G F (P0 ) G0
F (P1 )


F (A)


0

The kernel of F (P0 ) в†’ 0 is F (P0 ), so the 0th homology module (L0 F )(A) is F (P0 ) mod
the image of F (P1 ) в†’ F (P0 ) [=the kernel of F (P0 ) в†’ F (A).] By the п¬Ѓrst isomorphism
11

theorem and the right exactness of F , (L0 F )(A) в€ј F (A). To establish the other isomor-
=
phism, we switch to injective resolutions and reverse arrows:

G F (E0 ) G F (E1 )
0 y

F (A)
y

0

The kernel of F (E0 ) в†’ F (E1 ) is isomorphic to F (A) by left exactness of F , and the image
of 0 в†’ F (E0 ) is 0. Thus (R0 F )(A) в€ј F (A). в™Ј
=

S5.4 Lemma
If A is projective, then (Ln F )(A) = 0 for every n > 0; if A is injective, then (Rn F )(A) = 0
for every n > 0.

Proof. If A is projective [resp. injective], then 0 в†’ A в†’ A в†’ 0 is a projective [resp.
injective] resolution of A. Switching to a deleted resolution, we have 0 в†’ A в†’ 0 in each
case, and the result follows. в™Ј

S5.5 Deп¬Ѓnitions and Comments
If F is the right exact functor M вЉ—R , the left derived functor Ln F is called TorR (M, ).
n
If F is the left exact functor HomR (M, ), the right derived functor Rn F is called
Extn (M, ). It can be shown that the Ext functors can also be computed using pro-
R
jective resolutions and the contravariant hom functor. Speciп¬Ѓcally,

Extn (M, N ) = [Rn HomR ( , N )](M ).
R

A switch from injective to projective resolutions is a simpliп¬Ѓcation, because projective
resolutions are easier to п¬Ѓnd in practice.
The next three results sharpen Lemma S5.4. [The ring R is assumed п¬Ѓxed, and we
write вЉ—R simply as вЉ—. Similarly, we drop the R in TorR and ExtR . When discussing Tor,
we assume R commutative.]

S5.6 Proposition
If M is an R-module, the following conditions are equivalent.

(i) M is п¬‚at;
(ii) Torn (M, N ) = 0 for all n в‰Ґ 1 and all modules N ;
(iii) Tor1 (M, N ) = 0 for all modules N .
12

Proof. (i) implies (ii): Let Pв€— в†’ N be a projective resolution of N . Since M вЉ— is an
exact functor (see (10.8.1)), the sequence

В· В· В· в†’ M вЉ— P1 в†’ M вЉ— P0 в†’ M вЉ— N в†’ 0

is exact. Switching to a deleted resolution, we have exactness up to M вЉ— P1 but not at
M вЉ— P0 . Since the homology modules derived from an exact sequence are 0, the result
follows.
(ii) implies (iii): Take n = 1.
(iii) implies (i): If 0 в†’ A в†’ B в†’ C в†’ 0 is a short exact sequence, then by (S5.1), we
have the following long exact sequence:

В· В· В· Tor1 (M, C) в†’ Tor0 (M, A) в†’ Tor0 (M, B) в†’ Tor0 (M, C) в†’ 0.

By hypothesis, Tor1 (M, C) = 0, so by (S5.3),

0в†’M вЉ—Aв†’M вЉ—B в†’M вЉ—C в†’0

is exact, and therefore M is п¬‚at. в™Ј

S5.7 Proposition
If M is an R-module, the following conditions are equivalent.

(i) M is projective;
(ii) Extn (M, N ) = 0 for all n в‰Ґ 1 and all modules N ;
(iii) Ext1 (M, N ) = 0 for all modules N .

Proof. (i) implies (ii): By (S5.4) and (S5.5), Extn (M, N ) = [Extn ( , N )](M ) = 0 for
n в‰Ґ 1.
(ii) implies (iii): Take n = 1.
(iii) implies (i): Let 0 в†’ A в†’ B в†’ M в†’ 0 be a short exact sequence. If N is any
module, then using projective resolutions and the contravariant hom functor to construct
Ext, as in (S5.5), we get the following long exact sequence:

0 в†’ Ext0 (M, N ) в†’ Ext0 (B, N ) в†’ Ext0 (A, N ) в†’ Ext1 (M, N ) в†’ В· В· В·

By (iii) and (S5.3),

0 в†’ Hom(M, N ) в†’ Hom(B, N ) в†’ Hom(A, N ) в†’ 0

is exact. Take N = A and let g be the map from A to B. Then the map g в€— from Hom(B, A)
to Hom(A, A) is surjective. But 1A в€€ Hom(A, A), so there is a homomorphism f : B в†’ A
such that g в€— (f ) = f g = 1A . Therefore the sequence 0 в†’ A в†’ B в†’ M в†’ 0 splits, so by
(10.5.3), M is projective. в™Ј
13

S5.8 Corollary
If N is an R-module, the following conditions are equivalent.
(a) N is injective;
(b) Extn (M, N ) = 0 for all n в‰Ґ 1 and all modules M ;
(c) Ext1 (M, N ) = 0 for all modules M .
Proof. Simply saying вЂњdualityвЂќ may be unconvincing, so letвЂ™s give some details. For (a)
implies (b), we have [Extn (M, )](N ) = 0. For (c) implies (a), note that the exact
sequence 0 в†’ N в†’ A в†’ B в†’ 0 induces the exact sequence

0 в†’ Ext0 (M, N ) в†’ Ext0 (M, A) в†’ Ext0 (M, B) в†’ 0.

Replace Ext0 (M, N ) by Hom(M, N ), and similarly for the other terms. Then take M = B
and proceed exactly as in (S5.7). в™Ј

S6. Some Properties of Ext and Tor
We will compute Extn (A, B) and TorR (A, B) in several interesting cases.
R n

S6.1 Example
We will calculate Extn (Zm , B) for an arbitrary abelian group B. To ease the notational
Z
burden slightly, we will omit the subscript Z in Ext and Hom, and use = (most of the
time) when we really mean в€ј. We have the following projective resolution of Zm :
=

GZ GZ G Zm G0
m
0

where the m over the arrow indicates multiplication by m. Switching to a deleted resolu-
tion and applying the contravariant hom functor, we get

G Hom(Z, B) G Hom(Z, B) G0
m
0 y

Hom(Zm , B)

But by (9.4.1), we have

HomR (R, B) в€ј B (1)
=

and the above diagram becomes

GB GB G0
m
(2)
0

By (S5.3), Ext0 (Zm , B) = Hom(Zm , B). Now a homomorphism f from Zm to B is
determined by f (1), and f (m) = mf (1) = 0. If B(m) is the set of all elements of B
14

that are annihilated by m, that is, B(m) = {x в€€ B : mx = 0}, then the map of B(m) to
Hom(Zm , B) given by x в†’ f where f (1) = x, is an isomorphism. Thus

Ext0 (Zm , B) = B(m).

It follows from (2) that

Extn (Zm , B) = 0, n в‰Ґ 2

and

Ext1 (Zm , B) = ker(B в†’ 0)/ im(B в†’ B) = B/mB.

The computation for n в‰Ґ 2 is a special case of a more general result.

S6.2 Proposition
Extn (A, B) = 0 for all n в‰Ґ 2 and all abelian groups A and B.
Z

Proof. If B is embedded in an injective module E, we have the exact sequence

0 в†’ B в†’ E в†’ E/B в†’ 0.

This is an injective resolution of B since E/B is divisible, hence injective; see (10.6.5)
and (10.6.6). Applying the functor Hom(A, ) = HomZ (A, ) and switching to a deleted
resolution, we get the sequence

G Hom(A, E) G Hom(A, E/B) G0
0 y

Hom(A, B)

whose homology is 0 for all n в‰Ґ 2. в™Ј

S6.3 Lemma
Ext0 (Z, B) = HomZ (Z, B) = B and Ext1 (Z, B) = 0.
Z Z

Proof. The п¬Ѓrst equality follows from (S5.3) and the second from (1) of (S6.1). Since Z
is projective, the last statement follows from (S5.7). в™Ј

S6.4 Example
We will compute TorZ (Zm , B) for an arbitrary abelian group B. As before, we drop the
n
superscript Z and write = for в€ј. We use the same projective resolution of Zm as in (S6.1),
=
15

вЉ— B. Since R вЉ—R B в€ј B by (8.7.6), we reach diagram (2) as
and apply the functor =
before. Thus

Torn (Zm , B) = 0, n в‰Ґ 2;
Tor1 (Zm , B) = ker(B в†’ B) = {x в€€ B : mx = 0} = B(m);
Tor0 (Zm , B) = Zm вЉ— B = B/mB.

[To verify the last equality, use the universal mapping property of the tensor product to
produce a map of Zm вЉ— B to B/mB such that n вЉ— x в†’ n(x + mB). The inverse of this
map is x + mB в†’ 1 вЉ— x.]
The result for n в‰Ґ 2 generalizes as in (S6.2):

S6.5 Proposition
TorZ (A, B) = 0 for all n в‰Ґ 2 and all abelian groups A and B.
n

Proof. B is the homomorphic image of a free module F . If K is the kernel of the ho-
momorphism, then the exact sequence 0 в†’ K в†’ F в†’ B в†’ 0 is a free resolution of B.
[K is a submodule of a free module over a PID, hence is free.] Switching to a deleted
resolution and applying the tensor functor, we get a four term sequence as in (S6.2), and
the homology must be 0 for n в‰Ґ 2. в™Ј

S6.6 Lemma
Tor1 (Z, B) = Tor1 (A, Z) = 0; Tor0 (Z, B) = Z вЉ— B = B.

Proof. The п¬Ѓrst two equalities follow from (S5.6) since Z is п¬‚at. The other two equalities
follow from (S5.3) and (8.7.6). в™Ј

S6.7 Finitely generated abelian groups
We will show how to compute Extn (A, B) and Torn (A, B) for arbitrary п¬Ѓnitely generated
abelian groups A and B. By (4.6.3), A and B can be expressed as a п¬Ѓnite direct sum of
cyclic groups. Now Tor commutes with direct sums:

Torn (A, вЉ•r Bj ) = вЉ•r Torn (A, Bj ).
j=1 j=1

[The point is that if Pjв€— is a projective resolution of Bj , then the direct sum of the Pjв€—
is a projective resolution of вЉ•j Bj , by (10.5.4). Since the tensor functor is additive on
direct sums, by (8.8.6(b)), the Tor functor will be additive as well. Similar results hold
when the direct sum is in the п¬Ѓrst coordinate, and when Tor is replaced by Ext. (We use
(10.6.3) and Problems 5 and 6 of Section 10.9, and note that the direct product and the
direct sum are isomorphic when there are only п¬Ѓnitely many factors.)]
Thus to complete the computation, we need to know Ext(A, B) and Tor(A, B) when
A = Z or Zm and B = Z or Zn . We have already done most of the work. By (S6.2) and
16

(S6.5), Extn and Torn are identically 0 for n в‰Ґ 2. By (S6.1),

Ext0 (Zm , Z) = Z(m) = {x в€€ Z : mx = 0} = 0;
Ext0 (Zm , Zn ) = Zn (m) = {x в€€ Zn : mx = 0} = Zd

where d is the greatest common divisor of m and n. [For example, Z12 (8) = {0, 3, 6, 9} в€ј
=
Z4 . The point is that the product of two integers is their greatest common divisor times
their least common multiple.] By (S6.3),

Ext0 (Z, Z) = Hom(Z, Z) = Z; Ext0 (Z, Zn ) = Zn .

By (S6.1),

Ext1 (Zm , Z) = Z/mZ = Zm ;
Ext1 (Zm , Zn ) = Zn /mZn = Zd

as above. By (S5.7),

Ext1 (Z, Z) = Ext1 (Z, Zn ) = 0

By (S6.4),

Tor1 (Zm , Z) = Tor1 (Z, Zm ) = Z(m) = 0;
Tor1 (Zm , Zn ) = Zn (m) = Zd .

By (8.7.6) and (S6.4),

Tor0 (Z, Z) = Z;
Tor0 (Zm , Z) = Z/mZ = Zm ;
Tor0 (Zm , Zn ) = Zn /mZn = Zd .

Notice that Tor1 (A, B) is a torsion group for all п¬Ѓnitely generated abelian groups A and B.
This is a partial explanation of the term вЂњTorвЂќ. The Ext functor arises in the study of
group extensions.

S7. Base Change in the Tensor Product
Let M be an A-module, and suppose that we have a ring homomorphism from A to B (all
rings are assumed commutative). Then BвЉ—A M becomes a B module (hence an A-module)
via b(b вЉ— m) = bb вЉ— m. This is an example of base change, as discussed in (10.8.8). We
examine some frequently occurring cases. First, consider B = A/I, where I is an ideal
of A.

S7.1 Proposition
(A/I) вЉ—A M в€ј M/IM .
=
17

Proof. Apply the (right exact) tensor functor to the exact sequence of A-modules

0 в†’ I в†’ A в†’ A/I в†’ 0

to get the exact sequence

I вЉ—A M в†’ A вЉ—A M в†’ (A/I) вЉ—A M в†’ 0.

Recall from (8.7.6) that A вЉ—A M is isomorphic to M via a вЉ— m в†’ am. By the п¬Ѓrst
isomorphism theorem, (A/I) вЉ—A M is isomorphic to M mod the image of the map from
I вЉ—A M to M . This image is the collection of all п¬Ѓnite sums ai mi with ai в€€ I and
mi в€€ M , which is IM . в™Ј
Now consider B = S в€’1 A, where S is a multiplicative subset of A.

S7.2 Proposition
(S в€’1 A) вЉ—A M в€ј S в€’1 M .
=
Proof. The map from S в€’1 A Г— M to S в€’1 M given by (a/s, m) в†’ am/s is A-bilinear, so by
the universal mapping property of the tensor product, there is a linear map О± : S в€’1 A вЉ—A
M в†’ S в€’1 M such that О±((a/s) вЉ— m) = am/s. The inverse map ОІ is given by ОІ(m/s) =
(1/s) вЉ— m. To show that ОІ is well-deп¬Ѓned, suppose that m/s = m /s . Then for some
t в€€ S we have ts m = tsm . Thus

1/s вЉ— m = ts /tss вЉ— m = 1/tss вЉ— ts m = 1/tss вЉ— tsm = 1/s вЉ— m .

Now О± followed by ОІ takes a/sвЉ—m to am/s and then to 1/sвЉ—am = a/sвЉ—m. On the other
hand, ОІ followed by О± takes m/s to 1/s вЉ— m and then to m/s. Consequently, О± and ОІ are
inverses of each other and yield the desired isomorphism of S в€’1 A вЉ—A M and S в€’1 M . в™Ј

Finally, we look at B = A[X].

S7.3 Proposition

[X] вЉ—A M в€ј M [X]
=

where the elements of M [X] are of the form a0 m0 + a1 Xm1 + a2 X 2 m2 + В· В· В· + an X n mn ,
ai в€€ A, mi в€€ M , n = 0, 1, . . . .

Proof. This is very similar to (S7.2). In this case, the map О± from A[X] вЉ—A M to M [X]
takes f (X)вЉ—m to f (X)m, and the map ОІ from M [X] to A[X]вЉ—A M takes X i m to X i вЉ—m.
Here, there is no need to show that ОІ is well-deп¬Ѓned. в™Ј
Solutions Chapters 1вЂ“5

Section 1.1
1. Under multiplication, the positive integers form a monoid but not a group, and the
positive even integers form a semigroup but not a monoid.
2. With |a| denoting the order of a, we have |0| = 1, |1| = 6, |2| = 3, |3| = 2, |4| = 3,
and |5| = 6.
3. There is a subgroup of order 6/d for each divisor d of 6. We have Z6 itself (d = 1),
{0}(d = 6), {0, 2, 4}(d = 2), and {0, 3}(d = 3).
4. S forms a group under addition. The inverse operation is subtraction, and the zero
matrix is the additive identity.
5. S в€— does not form a group under multiplication, since a nonzero matrix whose deter-
minant is 0 does not have a multiplicative inverse.
6. If d is the smallest positive integer in H, then H consists of all multiples of d. For if
x в€€ H we have x = qd + r where 0 в‰¤ r < d. But then r = x в€’ qd в€€ H, so r must be
0.
7. Consider the rationals with addition mod 1, in other words identify rational numbers
that diп¬Ђer by an integer. Thus, for example, 1/3 = 4/3 = 7/3, etc. The group is
inп¬Ѓnite, but every element generates a п¬Ѓnite subgroup. For example, the subgroup
generated by 1/3 is {1/3, 2/3, 0}.
8. (ab)mn = (am )n (bn )m = 1, so the order of ab divides mn. Thus |ab| = m1 n1 where
m1 divides m and n1 divides n. Consequently,

am1 n1 bm1 n1 = 1 (1)

If m = m1 m2 , raise both sides of (1) to the power m2 to get bmn1 = 1. The order of b,
namely n, must divide mn1 , and since m and n are relatively prime, n must divide
n1 . But n1 divides n, hence n = n1 . Similarly, if n = n1 n2 we raise both sides of (1)
to the power n2 and conclude as above that m = m1 . But then |ab| = m1 n1 = mn,
as asserted.

If c belongs to both a and b then since c is a power of a and also a power of b, we
have cm = cn = 1. But then the order of c divides both m and n, and since m and n are
relatively prime, c has order 1, i.e., c = 1.

1
2

9. Let |a| = m, |b| = n. If [m, n] is the least common multiple, and (m, n) the greatest
common divisor, of m and n, then [m, n] = mn/(m, n). Examine the prime factoriza-
tions of m and n:
t
t
m = (pt1 В· В· В· pti )(pi+1 В· В· В· pjj ) = r r
i+1
1 i
u
u
n = (pu1 В· В· В· pui )(pi+1 В· В· В· pj j ) = s s
i+1
1 i

where tk в‰¤ uk for 1 в‰¤ k в‰¤ i, and tk в‰Ґ uk for i + 1 в‰¤ k в‰¤ j.
Now ar has order m/r and bs has order n/s, with m/r (= r ) and n/s (= s ) relatively
prime. By Problem 8, ar bs has order mn/rs = mn/(m, n) = [m, n]. Thus given
elements of orders m and n, we can construct another element whose order is the
least common multiple of m and n. Since the least common multiple of m, n and
q is [[m, n], q], we can inductively п¬Ѓnd an element whose order is the least common
multiple of the orders of all elements of G.
10. Choose an element a that belongs to H but not K, and an element b that belongs to
K but not H, where H and K are subgroups whose union is G. Then ab must belong
to either H or K, say ab = h в€€ H. But then b = aв€’1 h в€€ H, a contradiction. If
ab = k в€€ K, then a = kbв€’1 в€€ K, again a contradiction. To prove the last statement,
note that if H в€Є K is a subgroup, the п¬Ѓrst result with G replaced by H в€Є K implies
that H = H в€Є K or K = H в€Є K, in other words, K вЉ† H or H вЉ† K.
akm = 1 if and only if km is a multiple of n, and the smallest such multiple occurs
11.
when km is the least common multiple of n and k. Thus the order of ak is [n, k]/k.
Examination of the prime factorizations of n and k shows that [n, k]/k = n/(n, k).
We have x в€€ Ai iп¬Ђ x is a multiple of pi , and there are exactly n/pi multiples of pi
12.
between 1 and n. Similarly, x belongs to Ai в€© Aj iп¬Ђ x is divisible by pi pj , and there
are exactly pin j multiples of pi pj between 1 and n. The same technique works for all
p
other terms.
The set of positive integers in {1, 2, . . . , n} and not relatively prime to n is в€Єr Ai ,
13. i=1
so П•(n) = n в€’ | в€Єi=1 Ai |. By the principle of inclusion and exclusion from basic
r

combinatorics,
r r
|Ai | в€’ |Ai в€© Aj | + |Ai в€© Aj в€© Ak | в€’ В· В· В· + (в€’1)rв€’1 |A1 в€© A2 в€© В· В· В· Ar |.
Ai =
i=1 i=1 i<j i<j<k

By Problem 12,
r
1 1 1
П•(n) = n 1 в€’ в€’ + В· В· В· + (в€’1)r 1p1 p2 В· В· В· pr .
+
pi pi pj pi pj pk
i=1 i<j i<j<k

Thus П•(n) = n 1 в€’ p1 1 в€’ p2 В· В· В· 1 в€’ pr .
1 1 1

14. Let G be cyclic of prime order p. Since the only positive divisors of p are 1 and p, the
only subgroups of G are G and {1}.
15. No. Any non-identity element of G generates a cyclic subgroup H. If H вЉ‚ G, we
are п¬Ѓnished. If H = G, then G is isomorphic to the integers, and therefore has many
nontrivial proper subgroups. (See (1.1.4) and Problem 6 above.)
3

Section 1.2
1. The cycle decomposition is (1, 4)(2, 6, 5); there is one cycle of even length, so the
permutation is odd.
2. The elements are I, R = (A, B, C, D), R2 = (A, C)(B, D), R3 = (A, D, C, B),
F = (B, D), RF = (A, B)(C, D), R2 F = (A, C), R3 F = (A, D)(B, C).
3. Such a permutation can be written as (1, a1 , a2 , a3 , a4 ) where (a1 , a2 , a3 , a4 ) is a per-
mutation of {2, 3, 4, 5}. Thus the number of permutations is 4! = 24.
4. Select two symbols from 5, then two symbols from the remaining 3, and divide by 2
since, for example, (1, 4)(3, 5) is the same as (3, 5)(1, 4). The number of permutations
is 10(3)/2 = 15.
5. For example, (1, 2, 3)(1, 2) = (1, 3) but (1, 2)(1, 2, 3) = (2, 3).
6. We have V = {I, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)}. Thus V = {I, a, b, c} where
the product of any two distinct elements from {a, b, c} (in either order) is the third
element, and the square of each element is I. It follows that V is an abelian group.
7. This follows because the inverse of the cycle (a1 , a2 , . . . , ak ) is (ak , . . . , a2 , a1 ).
8. Pick 3 symbols out of 4 to be moved, then pick one of two possible orientations, e.g.,
(1, 2, 3) or (1, 3, 2). The number of 3-cycles in S4 is therefore 4(2) = 8.
9. If ПЂ is a 3-cycle, then ПЂ 3 = I, so ПЂ 4 = ПЂ. But ПЂ 4 = (ПЂ 2 )2 , and ПЂ 2 в€€ H by hypothesis,
so (ПЂ 2 )2 в€€ H because H is a group. Thus ПЂ в€€ H.
10. There are 5 inversions, 21, 41, 51, 43 and 53. Thus we have an odd number of
inversions and the permutation ПЂ = (1, 2, 4)(3, 5) is also odd.
11. This follows because a transposition of two adjacent symbols in the second row changes
the number of inversions by exactly 1. Therefore such a transposition changes the
parity of the number of inversions. Thus the parity of ПЂ coincides with the parity of the
number of inversions. In the given example, it takes 5 transpositions of adjacent digits
to bring 24513 into natural order 12345. It also takes 5 transpositions to create ПЂ:

ПЂ = (1, 5)(1, 4)(1, 2)(3, 5)(3, 4)

Section 1.3
1. If Ha = Hb then a = 1a = hb for some h в€€ H, so abв€’1 = h в€€ H. Conversely, if
abв€’1 = h в€€ H then Ha = Hhb = Hb.
2. Reп¬‚exivity: aaв€’1 = 1 в€€ H.
Symmetry: If abв€’1 в€€ H then (abв€’1 )в€’1 = baв€’1 в€€ H.
Transitivity: If abв€’1 в€€ H and bcв€’1 в€€ H then (abв€’1 )(bcв€’1 ) = acв€’1 в€€ H.
3. abв€’1 в€€ H iп¬Ђ (abв€’1 )в€’1 = baв€’1 в€€ H iп¬Ђ b в€€ Ha.
4. Haв€’1 = Hbв€’1 iп¬Ђ aв€’1 (bв€’1 )в€’1 = aв€’1 b в€€ H iп¬Ђ aH = bH.
5. Since a1 belongs to both aH and a1 H, we have a1 H = aH because the left cosets
partition G.
4

6. There are only two left cosets of H in G; one is H itself, and the other is, say, aH.
Similarly, there are only two right cosets, H and Hb. Since the left cosets partition G,
as do the right cosets, aH must coincide with Hb, so that every left coset if a right
coset.
7. The permutations on the list are e, (1, 2, 3), (1, 3, 2), (1, 2), (1, 3), and (2, 3), which
are in fact the 6 distinct permutations of {1, 2, 3}.
8. The left cosets of H are H = {e, b}, aH = {a, ab}, and a2 H = {a2 , a2 b}. The right
cosets of H are H = {e, b}, Ha = {a, ba} = {a, a2 b}, and Ha2 = {a2 , ba2 } = {a2 , ab}.
9. The computation of Problem 8 shows that the left cosets of H do not coincide with
the right cosets. Explicitly, aH and a2 H are not right cosets (and similarly, Ha and
Ha2 are not left cosets).
10. f (n) = f (1 + 1 + В· В· В· 1) = f (1) + f (1) + В· В· В· f (1) = r + r + В· В· В· r = rn.
11. In Problem 10, the image f (Z) must coincide with Z. But f (Z) consists of all multiples
of r, and the only way f (Z) can equal Z is for r to be В±1.
12. The automorphism group of Z is {I, в€’I} where (в€’I)2 = I. Thus the automorphisms
of Z form a cyclic group of order 2. (There is only one such group, up to isomorphism.)
13. Reп¬‚exivity: x = 1x1. Symmetry: If x = hyk, then y = hв€’1 xk в€’1 . Transitivity: if
x = h1 yk1 and y = h2 zk2 , then x = h1 h2 zk2 k1 .
14. HxK is the union over all k в€€ K of the right cosets H(xk), and also the union over
all h в€€ H of the left cosets (hx)K.

Section 1.4
1. Deп¬Ѓne f : Z в†’ Zn by f (x) = the residue class of x mod n. Then f is an epimorphism
with kernel nZ, and the result follows from the п¬Ѓrst isomorphism theorem.
2. Deп¬Ѓne f : Zn в†’ Zn/m by f (x) = x mod n/m. Then f is an epimorphism with
kernel Zm , and the result follows from the п¬Ѓrst isomorphism theorem. (In the concrete
example with n = 12, m = 4, we have f (0) = 0, f (1) = 1, f (2) = 2, f (3) = 0, f (4) = 1,
f (5) = 2, f (6) = 0, etc.)
3. f (xy) = axyaв€’1 = axaв€’1 ayaв€’1 = f (x)f (y), so f is a homomorphism. If b в€€ G, we
can solve axaв€’1 = b for x, namely x = aв€’1 ba, so f is surjective. If axaв€’1 = 1 then
ax = a, so x = 1 and f is injective. Thus f is an automorphism.
4. Note that fab (x) = abx(ab)в€’1 = a(bxbв€’1 )aв€’1 = fa (fb (x)), and y = fa (x) iп¬Ђ x =
faв€’1 (y), so that (fa )в€’1 = faв€’1 .
5. Deп¬Ѓne ОЁ : G в†’ Inn G, the group of inner automorphisms of G, by ОЁ(a) = fa . Then
ОЁ(ab) = fab = fa в—¦ fb = ОЁ(a)ОЁ(b), so ОЁ is a homomorphism (see the solution to
Problem 4). Since a is arbitrary, ОЁ is surjective. Now a belongs to ker ОЁ iп¬Ђ fa is the
identity function, i.e., axaв€’1 = x for all x в€€ G, in other words, a commutes with every
x in G. Thus ker ОЁ = Z(G), and the result follows from the п¬Ѓrst isomorphism theorem.
6. If f is an automorphism of Zn , then since 1 generates Zn , f is completely determined by
m = f (1), and since 1 has order n in Zn , m must have order n as well. But then m is a
unit mod n (see (1.1.5)), and f (r) = f (1 + 1 + В· В· В· 1) = f (1) + f (1) + В· В· В· f (1) = rf (1) =
5

rm. Conversely, any unit m mod n determines an automorphism Оё(m) = multiplication
by m. The correspondence between m and Оё(m) is a group isomorphism because
Оё(m1 m2 ) = Оё(m1 ) в—¦ Оё(m2 ).
7. The п¬Ѓrst assertion follows from the observation that HN is the subgroup generated
by H в€Є N (see (1.3.6)). For the second assertion, note that if K is a subgroup of G
contained in both H and N , then K is contained in H в€© N .
8. If g(x) = y, then g в—¦ fa в—¦ g в€’1 maps y to g(axaв€’1 ) = g(a)y[g(a)]в€’1 .
9. If G is abelian, then fa (x) = axaв€’1 = aaв€’1 x = x.

Section 1.5
1. C2 Г— C2 has 4 elements 1 = (1, 1), О± = (a, 1), ОІ = (1, a) and Оі = (a, a), and the
product of any two distinct elements from {О±, ОІ, Оі} is the third. Since each of О±, ОІ,
Оі has order 2 (and 1 has order 1), there is no element of order 4 and C2 Г— C2 is not
cyclic.
2. The four group is V = {I, a, b, c} where the product of any two distinct elements from
{a, b, c} is the third. Therefore, the correspondence 1 в†’ I, О± в†’ a, ОІ в†’ b, Оі в†’ c is an
isomorphism of C2 Г— C2 and V .
3. Let C2 = {1, a} with a2 = 1, and C3 = {1, b, b2 } with b3 = 1. Then (a, b) generates
C2 Г— C3 , since the successive powers of this element are (a, b), (1, b2 ), (a, 1), (1, b),
(a, b2 ), and (1, 1). Therefore C2 Г— C3 is cyclic of order 6, i.e., isomorphic to C6 .
4. Proceed as in Problem 3. If a has order n in Cn and b has order m in Cm , then (a, b)
has order nm in Cn Г— Cm , so that Cn Г— Cm is cyclic of order nm.
5. Suppose that (a, b) is a generator of the cyclic group Cn Г— Cm . Then a must generate
Cn and b must generate Cm (recall that Cn Г— {1} can be identiп¬Ѓed with Cn ). But
(a, b)k = 1 iп¬Ђ ak = bk = 1, and it follows that the order of (a, b) is the least common
multiple of the orders of a and b, i.e., the least common multiple of n and m. Since n
and m are not relatively prime, the least common multiple is strictly smaller than nm,
so that (a, b) cannot possibly generate Cn Г— Cm , a contradiction.
6. By (1.3.3), G and H are both cyclic. Since p and q are distinct primes, they are
relatively prime, and by Problem 4, G Г— H is cyclic.
7. Deп¬Ѓne f : H Г— K в†’ K Г— H by f (h, k) = (k, h). It follows from the deп¬Ѓnition of direct
product that f is an isomorphism.
8. Deп¬Ѓne f1 : G Г— H Г— K в†’ G Г— (H Г— K) by f1 (g, h, k) = (g, (h, k)), and deп¬Ѓne f2 : G Г—
H Г— K в†’ (G Г— H) Г— K by f2 (g, h, k) = ((g, h), k). It follows from the deп¬Ѓnition of
direct product that f1 and f2 are isomorphisms.

Section 2.1
1. Never. If f is a polynomial whose degree is at least 1, then f cannot have an inverse.
For if f (X)g(X) = 1, then the leading coeп¬ѓcient of g would have to be 0, which is
impossible.
6

2. If f (X)g(X) = 1, then (see Problem 1) f and g are polynomials of degree 0, in other
words, elements of R. Thus the units of R[X] are simply the nonzero elements of R.
3. (a) No element of the form a1 X + a2 X 2 + В· В· В· can have an inverse.
(b) For example, 1 в€’ X is a unit because (1 в€’ X)(1 + X + X 2 + X 3 + В· В· В· ) = 1.
4. Since Z[i] is a subset of the п¬Ѓeld C of complex numbers, there can be no zero divisors
in Z[i]. If w is a nonzero Gaussian integer, then w has an inverse in C, but the inverse
need not belong to Z[i]. For example, (1 + i)в€’1 = 1 в€’ 1 i.
2 2
5. If z = a + bi with a and b integers, then |z|2 = a2 + b2 , so that if z is not zero, we
must have |z| в‰Ґ 1. Thus if zw = 1, so that |z||w| = 1, we have |z| = 1, and the only
possibilities are a = 0, b = В±1 or a = В±1, b = 0. Consequently, the units of Z[i] are 1,
в€’1, i and в€’i.
6. All identities follow directly from the deп¬Ѓnition of multiplication of quaternions. Al-
ternatively, (b) can be deduced from (a) by interchanging x1 and x2 , y1 and y2 , z1
and z2 , and w1 and w2 . Then the second identity of (c) can be deduced by noting
that the quaternion on the right side of the equals sign in (a) is the conjugate of the
quaternion on the right side of the equals sign in (b).
7. Multiply identities (a) and (b), and use (c). (This is not how Euler discovered the
identity; quaternions were not invented until much later.)
8. The veriп¬Ѓcation that End G is an abelian group under addition uses the fact that G is
an abelian group. The additive identity is the zero function, and the additive inverse
of f is given by (в€’f )(a) = в€’f (a). Multiplication is associative because composition
of functions is associative. To establish the distributive laws, note that the value of
(f + g)h at the element a в€€ G is f (h(a)) + g(h(a)), so that (f + g)h = f h + gh.
Furthermore, the value of f (g + h) at a is f (g(a) + h(a)) = f (g(a)) + f (h(a)) since
f is an endomorphism. Therefore f (g + h) = f h + gh. The multiplicative identity is
the identity function, given by E(a) = a for all a.
9. An endomorphism that has an inverse must be an isomorphism of G with itself. Thus
the units of the ring End G are the automorphisms of G.
10. Use EulerвЂ™s identity with x1 = 1, y1 = 2, z1 = 2, w1 = 5 (34 = 12 + 22 + 22 + 52 )
and x2 = 1, y2 = 1, z2 = 4, w2 = 6 (54 = 12 + 12 + 42 + 62 ). The result is 1836 =
(34)(54) = 412 + 92 + 52 + 72 . The decomposition is not unique; another possibility
is x1 = 0, y1 = 0, z1 = 3, w1 = 5, x2 = 0, y2 = 1, z2 = 2, w2 = 7.
11. In all four cases, sums and products of matrices of the given type are also of that
type. But in (b), there is no matrix of the given form that can serve as the identity..
Thus the sets (a), (c) and (d) are rings, but (b) is not.

Section 2.2
1. By Section 1.1, Problem 6, the additive subgroups of Z are of the form (n) = all
multiples of n. But if x в€€ (n) and r в€€ Z then rx в€€ (n), so each (n) is an ideal as well.
2. If the n by n matrix A is 0 except perhaps in column k, and B is any n by n matrix,
then BA is 0 except perhaps in column k. Similarly, if A is 0 oп¬Ђ row k, then so is AB.
7

3. (a) This follows from the deп¬Ѓnition of matrix multiplication.
(b) In (a) we have ajr = 0 for r = k, and the result follows.
(c) By (b), the ith term of the sum is a matrix with cik in the ik position, and 0вЂ™s
elsewhere. The sum therefore coincides with C.
4. The statement about left ideals follows from the formula of Problem 3(c). The result
for right ideals is proved in a similar fashion. Explicitly, AEij has column i of A as its
j th column, with 0вЂ™s elsewhere. If A в€€ Rk then AEij has aki in the kj position, with
0вЂ™s elsewhere, so if aki = 0 we have AEij aв€’1 = Ekj . Thus if C в€€ Rk then
ki

n
AEij aв€’1 ckj = C.
ki
j=1

5. If I is a two-sided ideal and A в€€ I with ars = 0, then by considering products of the
form aв€’1 Epq AEkl (which have the eп¬Ђect of selecting an entry of A and sliding it from
rs
one row or column to another), we can show that every matrix Eij belongs to I. Since
every matrix is a linear combination of the Eij , it follows that I = Mn (R).
6. A polynomial with no constant term is of the form a1 X + a2 X 2 + В· В· В· an X n = Xg(X).
Conversely, a polynomial expressible as Xg(X) has no constant term. Thus we may
take f = X.
7. Let a be a nonzero element of R. Then the principal ideal (a) is not {0}, so (a) = R.
Thus 1 в€€ (a), so there is an element b в€€ R such that ab = 1.
8. Since an ideal I is a п¬Ѓnite set in this case, it must have a п¬Ѓnite set of generators
x1 , . . . , xk . Let d be the greatest common divisor of the xi . Every element of I is
of the form a1 x1 + В· В· В· + ak xk , and hence is a multiple of d. Thus I вЉ† (d). But
d в€€ I, because there are integers ai such that i ai xi = d. Consequently, (d) вЉ† I.
[Technically, arithmetic is modulo n, but we get around this diп¬ѓculty by noting that
if ab = c as integers, then ab в‰Ў c modulo n.]

Section 2.3
1. Use the same maps as before, and apply the п¬Ѓrst isomorphism theorem for rings.
2. If In is the set of multiples of n > 1 in the ring of integers, then In is an ideal but not
a subring (since 1 в€€ In ). Z is a subring of the rational numbers Q but not an ideal,
/
since a rational number times an integer need not be an integer.
3. In parts (2) and (3) of the Chinese remainder theorem, take R = Z and Ii = the set of
multiples of mi .
4. Apply part (4) of the Chinese remainder theorem with R = Z and Ii = the set of
multiples of mi .
5. To prove the п¬Ѓrst statement, deп¬Ѓne f : R в†’ R2 by f (r1 , r2 ) = r2 . Then f is a ring
homomorphism with kernel R1 and image R2 . By the п¬Ѓrst isomorphism theorem for
rings, R/R1 в€ј R2 . A symmetrical argument proves the second statement. In practice,
=
we tend to forget about the primes and write R/R1 в€ј R2 and R/R2 в€ј R1 . There is
= =
8

also a tendency to identify a ring with its isomorphic copy, and write R/R1 = R2 and
R/R2 = R1 This should not cause any diп¬ѓculty if you add, mentally at least, вЂќup to
isomorphismвЂќ.
6. The product is always a subset of the intersection, by deп¬Ѓnition. First consider the
case of two ideals. Then 1 = a1 + a2 for some a1 в€€ I1 , a2 в€€ I2 . If b в€€ I1 в€© I2 ,
then b = b1 = ba1 + ba2 в€€ I1 I2 . The case of more than two ideals is handled by
induction. Note that R = (I1 + In )(I2 + In ) В· В· В· (Inв€’1 + In ) вЉ† (I1 В· В· В· Inв€’1 ) + In .
Therefore (I1 В· В· В· Inв€’1 ) + In = R. By the n = 2 case and the induction hypothesis,
I1 В· В· В· Inв€’1 In = (I1 В· В· В· Inв€’1 ) в€© In = I1 в€© I2 в€© В· В· В· в€© In .
7. Let a + в€©i Ii map to (1 + I1 , 0 + I2 , c3 + I3 , . . . , cn + In ), where the cj are arbitrary.
Then 1 в€’ a в€€ I1 and a в€€ I2 , so 1 = (1 в€’ a) + a в€€ I1 + I2 . Thus I1 + I2 = R, and
similarly Ii + Ij = R for all i = j.

Section 2.4
1. If n = rs with r, s > 1 then r в€€ n , s в€€ n , but rs в€€ n , so that n is not prime. But
/ /
Z/ n is isomorphic to Zn , the ring of integers modulo n (see Section 2.3, Problem 1).
If n is prime, then Zn is a п¬Ѓeld, in particular an integral domain, hence n is a prime
ideal by (2.4.5).
2. By Problem 1, I is of the form p where p is prime. Since Z/ p is isomorphic to Zp ,
which is a п¬Ѓeld, p is maximal by (2.4.3).
3. The epimorphism a0 + a1 X + a2 X 2 + В· В· В· в†’ a0 of F [[X]] onto F has kernel X , and
the result follows from (2.4.7).
4. The ideal I = 2, X is not proper; since 2 в€€ I and 1 в€€ F вЉ† F [[X]], we have 1 в€€ I and
2
therefore I = F [[X]]. The key point is that F is a п¬Ѓeld, whereas Z is not.
5. Suppose that f (X) = a0 + a1 X + В· В· В· belongs to I but not to X . Then a0 cannot
be 0, so by ordinary long division we can п¬Ѓnd g(X) в€€ F [[X]] such that f (X)g(X) = 1.
But then 1 в€€ I, contradicting the assumption that I is proper.
6. Let f (X) = an X n + an+1 X n+1 + В· В· В· , an = 0, be an element of the ideal I, with n
as small as possible. Then f (X) в€€ (X n ), and if g(X) is any element of I, we have
g(X) в€€ (X m ) for some m в‰Ґ n. Thus I вЉ† (X n ). Conversely, if f (X) = X n g(X) в€€ I,
with g(X) = an + an+1 X + В· В· В· , an = 0,, then as in Problem 5, g(X) is a unit, and
therefore X n в€€ I. Thus (X n ) вЉ† I, so that I = (X n ), as claimed.
7. f в€’1 (P ) is an additive subgroup by (1.3.15), part (ii). If a в€€ f в€’1 (P ) and r в€€ R, then
f (ra) = f (r)f (a) в€€ P , so ra в€€ f в€’1 (P ). Thus f в€’1 (P ) is an ideal. If ab в€€ f в€’1 (P ), then
f (a)f (b) в€€ P , so either a or b must belong to f в€’1 (P ). If f в€’1 (P ) = R, then f maps
eveything in R, including 1, into P ; thus f в€’1 (P ) is proper. (Another method: As a
proper ideal, P is the kernel of some ring homomorphism ПЂ. Consequently, f в€’1 (P ) is
the kernel of ПЂ в—¦ f , which is also a ring homomorphism. Therefore f в€’1 (P ) is a proper
ideal.) Consequently, f в€’1 (P ) is prime.
8. Let S be a п¬Ѓeld, and R an integral domain contained in S, and assume that R is not
a п¬Ѓeld. For example, let R = Z, S = Q. Take f to be the inclusion map. Then {0} is
a maximal ideal of S, but f в€’1 ({0}) = {0} is a prime but not maximal ideal of R.
9

9. If P = I в€©J with P вЉ‚ I and P вЉ‚ J, choose a в€€ I \P and b в€€ J \P . Then ab в€€ I в€©J = P ,
contradicting the assumption that P is prime.

Section 2.5
1. Any number that divides a and b divides b and r1 , and conversely, any number that
divides b and r1 divides a and b. Iterating this argument, we п¬Ѓnd that gcd(a, b) =
gcd(b, r1 ) = gcd(r1 , r2 ) = В· В· В· = gcd(rjв€’1 , rj ) = rj .
2. This follows by successive substitution. We start with rj = rjв€’2 в€’ rjв€’1 qj , continue
with rjв€’1 = rjв€’3 в€’ rjв€’2 qjв€’1 , rjв€’2 = rjв€’4 в€’ rjв€’3 qjв€’2 , and proceed up the ladder until
we have expressed d as a linear combination of a and b. There is an easier way, as
Problem 3 shows.
3. The п¬Ѓrst equation of the three describes the steps of the algorithm. We wish to prove
that axi + byi = ri , that is,

a(xiв€’2 в€’ qi xiв€’1 ) + b(yiв€’2 в€’ qi yiв€’1 ) = ri . (1)

But this follows by induction: if axiв€’2 + byiв€’2 = riв€’2 and axiв€’1 + byiв€’1 = riв€’1 , then
the left side of (1) is riв€’2 в€’qi riв€’1 , which is ri by deп¬Ѓnition of the Euclidean algorithm.
4. We have the following table:

i qi+1 ri xi yi
в€’1 вЂ” 123 1 0
0 2 54 0 1
в€’2
1 3 15 1
в€’3
2 1 9 7
в€’9
3 1 6 4
в€’7
4 2 3 16

For example, to go from i = 1 to i = 2 we have x2 = x0 в€’ q2 x1 = 0 в€’ 3(1) = в€’3,
y2 = y0 в€’ q2 y1 = 1 в€’ 3(в€’2) = 7, and r2 = r0 в€’ q2 r1 = 54 в€’ 3(15) = 9; also,
q3 = 15/9 = 1. We have ax2 + by2 = 123(в€’3) + 54(7) = 9 = r2 , as expected. The
process terminates with 123(в€’7) + 54(16) = 3 = d.
5. If p is composite, say p = rs with 1 < r < p, 1 < s < p, then rs is 0 in Zp but r
and s are nonzero, so Zp is not a п¬Ѓeld. If p is prime and a is not zero in Zp then the
greatest common divisor of a and p is 1, and consequently there are integers x and
y such that ax + py = 1. In Zp this becomes ax = 1, so that every nonzero element
in Zp has an inverse in Zp , proving that Zp is a п¬Ѓeld.
6. Since f (X) and g(X) are multiples of d(X), so are all linear combinations a(X)f (X)+
b(X)g(X), and consequently I вЉ† J. By Problem 2, there are polynomials a(X) and
b(X) such that a(X)f (X) + b(X)g(X) = d(X), so that d(X) belongs to I. Since I is
an ideal, every multiple of d(X) belongs to I, and therefore J вЉ† I.
n
7. Take f (X) = i=0 bi Pi (X).
10

8. If g(X) is another polynomial such that g(ai ) = f (ai ) for all i, then f and g agree at
n+1 points, so that f (X)в€’g(X) has more than n roots in F . By (2.5.3), f (X)в€’g(X)
must be the zero polynomial.
9. If F has only п¬Ѓnitely many elements a1 , . . . , an , take f (X) = (X в€’ a1 ) В· В· В· (X в€’ an ).
10. Let F be the complex numbers C. Then every polynomial of degree n has exactly n
roots, counting multiplicity. Thus if f (a) = 0 at more than n points a, in particular
if f vanishes at every point of C, then f = 0. More generally, F can be any inп¬Ѓnite
п¬Ѓeld (use (2.5.3)).

Section 2.6
1. If r = 0 then I contains a unit, so that 1 в€€ I and I = R.
2. If b в€€ p1 then b + p1 = p1 , so b + p1 has an inverse in R/ p1 , say c + p1 . Thus
/
(b + p1 )(c + p1 ) = 1 + p1 , hence (bc в€’ 1) + p1 = p1 , so bc в€’ 1 в€€ p1 .
3. If bc в€’ dp1 = 1 then bcp2 В· В· В· pn в€’ dp1 В· В· В· pn = p2 В· В· В· pn , and since b and p1 В· В· В· pn belong
to I, so does p2 В· В· В· pn , contradicting the minimality of n. (If n = 1, then 1 в€€ I, so
I = R.)
4. If a, b в€€ R and x, y в€€ J then (ax + by)p1 = xp1 a + yp1 b. Since x, y в€€ J we have xp1 в€€ I
and yp1 в€€ I, so that (ax + by)p1 в€€ I, hence ax + by в€€ J.
5. If x в€€ J then xp1 в€€ I, so Jp1 вЉ† I. Now I вЉ† p1 by Problem 3, so if a в€€ I then
a = xp1 for some x в€€ R. But then x в€€ J by deп¬Ѓnition of J, so a = xp1 в€€ Jp1 .
6. Since J contains a product of fewer than n primes, J is principal by the induction
hypothesis. If J = d then by Problem 5, I = J p1 . But then I = dp1 , and the
result follows. (If n = 1, then p1 в€€ I, hence 1 в€€ J , so J = R and I = J p1 = p1 .)
7. Assume that P вЉ† Q. Then p = aq for some a в€€ R, so aq в€€ P . Since P is prime, either
a or q belongs to P . In the second case, Q вЉ† P and we are п¬Ѓnished. Thus assume
a в€€ P , so that a = bp for some b в€€ R. Then p = aq = bpq, and since R is an integral
domain and p = 0, we have bq = 1, so q is a unit and Q = R, a contradiction of the
assumption that Q is prime.
8. Let x be a nonzero element of P , with x = up1 В· В· В· pn , u a unit and the pi prime. Then
p1 В· В· В· pn = uв€’1 x в€€ P , and since P is prime, some pi belongs to P . Thus P contains
the nonzero principal prime ideal pi .

Section 2.7
1. If m is a generator of the indicated ideal then m belongs to all ai , so each ai divides
m. If each ai divides b then b is in every ai , so b в€€ в€©n ai = m , so m divides b.
i=1
Thus m is a least common multiple of A. Now suppose that m is an lcm of A, and let
в€©n ai = c . Then c belongs to every ai , so each ai divides c. Since m = lcm(A),
i=1
m divides c, so c is a subset of m . But again since m = lcm(A), each ai divides m,
so m в€€ в€©n ai = c . Therefore m вЉ† c , hence m = c , and m is a generator
i=1
of в€©n ai .
i=1
11

2. Let a = 11 + 3i, b = 8 в€’ i. Then a/b = (11 + 3i)(8 + i)/65 = 85/65 + i35/65. Thus
we may take x0 = y0 = 1, and the п¬Ѓrst quotient is q1 = 1 + i. The п¬Ѓrst remainder
is r1 = a в€’ bq1 = (11 + 3i) в€’ (8 в€’ i)(1 + i) = 2 в€’ 4i. The next step in the Euclidean
algorithm is (8 в€’ i)/(2 в€’ 4i) = (8 в€’ i)(2 + 4i)/20 = 1 + (3i/2). Thus the second quotient
is q2 = 1 + i (q2 = 1 + 2i would be equally good). The second remainder is r2 =
(8в€’i)в€’(2в€’4i)(1+i) = 2+i. The next step is (2в€’4i)/(2+i) = (2в€’4i)(2в€’i)/5 = в€’2i,
so q3 = в€’2i, r3 = 0. The gcd is the last divisor, namely 2 + i.
3. We have ОЁ(1) в‰¤ ОЁ(1(a)) = ОЁ(a) for every nonzero a. If a is a unit with ab = 1,
then ОЁ(a) в‰¤ ОЁ(ab) = ОЁ(1), so ОЁ(a) = ОЁ(1). Conversely, suppose that a = 0 and
ОЁ(a) = ОЁ(1). Divide 1 by a to get 1 = aq + r, where r = 0 or ОЁ(r) < ОЁ(a) = ОЁ(1).
But if r = 0 then ОЁ(r) must be greater than or equal to ОЁ(1), so we must have r = 0.
Therefore 1 = aq, and a is a unit.
в€љ в€љ
4. ОЁ((a1 + b1 d)(a2 + b2 d)) в€љ
= П€(a1 a2 + b1 b2 d + (a1 b2 + a2 b1 ) d)
в€љ в€љ
= a1 a2 + b1 b2 d + (a1 b2 + a2 b1 ) d a1 a2 + b1 b2 d в€’ (a1 b2 + a2 b1 ) d ;
в€љ в€љ
ОЁ(a1 + b1 d)ОЁ(a2 + b2 d) в€љ в€љ в€љ в€љ
= a1 + b1 d a2 + b2 d a1 в€’ b1 d a2 в€’ b2 d
and it follows that ОЁ(О±ОІ) = ОЁ(О±)ОЁ(ОІ). Now ОЁ(О±) в‰Ґ 1 for all nonzero О±, for if
ОЁ(О±) = |a2 в€’ db2 | = 0, then a2 = db2 . But if b = 0 then d = (a/b)2 , contradicting the
assumption that d is not a perfect square. Thus b = 0, so a is 0 as well, and О± = 0, a
contradiction. Thus ОЁ(О±ОІ) = ОЁ(О±)ОЁ(ОІ) в‰Ґ ОЁ(О±).
в€љ в€љ
5. Either d or d в€’ 1 is в€љ divides d(d в€’ 1) = d2 в€’ d = (d + d)(d в€’ d). But 2
even, so 2 в€љ в€љ в€љ
does not divide d + d or в€љ в€’ d. For example, if 2(a + b d) = d + d) for integers
d в€љ в€љ
a, b then 2a в€’ d = (1 в€’ 2b) d, which is impossible since d is irrational. (If d = r/s
then r2 = ds2 , which cannot happen if d is not a perfect square.)
6. Deп¬Ѓne ОЁ as inв€љ Problem 4 (and Example (2.7.5)). Suppose 2 = О±ОІ where О± and ОІ are
nonunits in Z[ d]. Then 4 = ОЁ(2) = ОЁ(О±)ОЁ(ОІ), with ОЁ(О±), ОЁ(ОІ) > 1 by Problems 3
в€љ
and 4. But then ОЁ(О±) = ОЁ(ОІ) = 2. If О± = a + b d then |a2 в€’ db2 | = 2, so a2 в€’ db2 is
either 2 or в€’2. Therefore if b = 0 (so that b2 в‰Ґ 1), then since d в‰¤ в€’3 we have

a2 в€’ db2 в‰Ґ 0 + 3(1) = 3,

a contradiction. Thus b = 0, so О± = a, and 2 = ОЁ(a) = a2 , an impossibility for a в€€ Z.
7. This follows from Problems 5 and 6, along with (2.6.4).
8. Just as with ordinary integers, the product of two Gaussian integers is their greatest
common divisor times their least common multiple. Thus by Problem 2, the lcm is
(11 + 3i)(8 в€’ i)/(2 + i) = 39 в€’ 13i.
9. If О± = ОІОі, then ОЁ(О±) = ОЁ(ОІ)ОЁ(Оі). By hypothesis, either ОЁ(ОІ) or ОЁ(Оі) is 1(= ОЁ(1)).
By Problem 3, either ОІ or Оі is a unit.

Section 2.8
1. If D is a п¬Ѓeld, then the quotient п¬Ѓeld F , which can be viewed as the smallest п¬Ѓeld
containing D, is D itself. Strictly speaking, F is isomorphic to D; the embedding map
12

f (a) = a/1 is surjective, hence an isomorphism. To see this, note that if a/b в€€ F , then
a/b = abв€’1 /1 = f (abв€’1 ).
2. The quotient п¬Ѓeld consists of all rational functions f (X)/g(X), where f (X) and g(X)
are polynomials in F [X] and g(X) is not the zero polynomial. To see this, note that
the collection of rational functions is in fact a п¬Ѓeld, and any п¬Ѓeld containing F [X] must
contain all such rational functions.
a c e a c e adf + bcf + bde
3. + + and + + both compute to be .
b df b d f bdf
ac e a cf + de acf + ade
4. + = = and
bd f b df bdf
ac ae acbf + bdae acf + dae acf + ade
+ = = = .
b2 df
bd bf bdf bdf
5. If g is any extension of h and a/b в€€ F , there is only one possible choice for g(a/b),
namely h(a)/h(b). (Since b = 0 and h is a monomorphism, h(b) = 0.) If we deп¬Ѓne g
this way, then g(a) = g(a/1) = h(a)/h(1) = h(a), so g is in fact an extension of f .
Furthermore, if a/b = c/d then since h is a monomorphism, h(a)/h(b) = h(c)/h(d).
Therefore g is well-deп¬Ѓned. Again since h is a monomorphism, it follows that g a + d =c
b
g a + g d and g a d = g a g d . Since g is an extension of h, we have g(1) = 1,
c c c
b b b
so g is a homomorphism. Finally, if g(a/b) = 0, then h(a) = 0, so a = 0 by injectivity
of h. Thus g is a monomorphism.
6. The problem is that h is not injective. As before, if g is to be an extension of h, we
must have g(a/b) = h(a)/h(b). But if b is a multiple of p, then h(b) is zero, so no
such g can exist.
7. We must have g(a/b) = g(a/1)g((b/1)в€’1 ) = g(a)g(b)в€’1 .
8. If a/b = c/d, then for some s в€€ S we have s(adв€’bc) = 0. So g(s)[g(a)g(d)в€’g(b)g(c)] = 0.
Since g(s) is a unit, we may multiply by its inverse to get g(a)g(d) = g(b)g(c), hence
g(a)g(b)в€’1 = g(c)g(d)в€’1 , proving that g is well-deп¬Ѓned. To show that g is a homomor-
phism, we compute
a c ad + bc
= g(ad + bc)g(bd)в€’1
g + =g
b d bd
a c
= [g(a)g(d) + g(b)g(c)]g(b)в€’1 g(d)в€’1 = g +g
b d
ac a c
Similarly, we have g =g g and g(1) = 1.
bd b d

Section 2.9
1. We have an (u/v)n + anв€’1 (u/v)nв€’1 + В· В· В· + a1 (u/v) + a0 = 0; multiply by v n to get

an un + anв€’1 unв€’1 v + В· В· В· + a1 uv nв€’1 + a0 v n = 0.

Therefore

an un = в€’anв€’1 unв€’1 v в€’ В· В· В· в€’ a1 uv nв€’1 в€’ a0 v n .
13

Since v divides the right side of this equation, it must divide the left side as well, and
since u and v are relatively prime, v must divide an . Similarly,

a0 v n = в€’an un в€’ anв€’1 unв€’1 v в€’ В· В· В· в€’ a1 uv nв€’1 ,

so u divides a0 .
2. X n в€’ p satisп¬Ѓes EisensteinвЂ™s criterion, and since the polynomial is primitive, it is
irreducible over Z.
3. f3 (X) = X 3 +2X +1, which is irreducible over Z3 . For if f3 (X) were reducible over Z3 ,
it would have a linear factor (since it is a cubic), necessarily X в€’ 1 or X + 1(= X в€’ 2).
But then 1 or 2 would be a root of f3 , a contradiction since f3 (1) = 1 and f3 (2) = 1
(mod 3).
4. By Eisenstein, X 4 + 3 is irreducible over Z. The substitution X = Y + 1 yields
Y 4 + 4Y 3 + 6Y 2 + 4Y + 4, which is therefore irreducible in Z[Y ]. Thus X 4 + 4X 3 +
6X 2 + 4X + 4 is irreducible in Z[X], i.e., irreducible over Z.
5. Note that n, X is a proper ideal since it cannot contain 1. If n, X = f then
n в€€ f , so n is a multiple of f . Thus f is constant (= 1), in which case X в€€ f .
/
6. Since 1 в€€ X, Y , X, Y is a proper ideal. Suppose X, Y = f . Then Y is a
/
multiple of f , so f is a polynomial in Y alone (in fact f = cY ). But then X в€€ f , a
/
7. If p = X + i, then p is irreducible since X + i is of degree 1. Furthermore, p divides
X 2 + 1 but p2 does not. Take the ring R to be C[X, Y ] = (C[X])[Y ] and apply
EisensteinвЂ™s criterion.
8. Write f (X, Y ) as Y 3 +(X 3 +1) and take p = X +1. Since X 3 +1 = (X +1)(X 2 в€’X +1)
and X + 1 does not divide X 2 в€’ X + 1, the result follows as in Problem 7.

Section 3.1
1. F (S) consists of all quotients of п¬Ѓnite linear combinations (with coeп¬ѓcients in F ) of
п¬Ѓnite products of elements of S. To prove this, note п¬Ѓrst that the set A of all such
quotients is a п¬Ѓeld. Then observe that any п¬Ѓeld containing F and S must contain
A, in particular, A вЉ† F (S). But F (S) is the smallest subп¬Ѓeld containing F and S,
so F (S) вЉ† A.
2. The composite consists of all quotients of п¬Ѓnite sums of products of the form xi1 xi2 В· В· В·
xin , n = 1, 2, . . . , where i1 , i2 , . . . , in в€€ I and xij в€€ Kij . As in Problem 1, the set A
of all such quotients is a п¬Ѓeld, and any п¬Ѓeld that contains all the Ki must contain A.
3. By (3.1.9), [F [О±] : F ] = [F [О±] : F [ОІ]][F [ОІ] : F ], and since the degree of any extension
is at least 1, the result follows.
в€љ
4. Let min(в€’1 + 2, Q) = a0 + a1 X + X 2 в€љ polynomialв€љ degree 1 cannot work в€љ
(a of because
в€љ
в€’1 +в€љ 2 в€€ Q). Then a0 + a1 (в€’1 + 2) + (в€’1 + 2) = 0. Since (в€’1 + 2)2 =
2
/
3 в€’ 2 2, we have a0 в€’ a1 + 3 = 0 and a1 в€’ 2 = 0, so a0 = в€’1, a1 = 2. Therefore
в€љ
min(в€’1 + 2 2, Q) = X 2 + 2X в€’ 1.
14

5. Let ОІ = b0 + b1 О± + В· В· В· + bnв€’1 О±nв€’1 . Then for some a0 , . . . , an в€€ F we have a0 + a1 ОІ +
В· В· В· + an ОІ n = 0. Substituting the expression for ОІ in terms of О± into this equation,
reducing to a polynomial in О± of degree at most n в€’ 1 (as in the proof of (3.1.7)), and
setting the coeп¬ѓcients of the О±i , i = 0, 1, . . . , n в€’ 1 equal to zero (remember that the
О±i form a basis for F [О±] over F ), we get n linear equations in the n + 1 unknowns ai ,
i = 0, . . . , n. We know that a solution exists because ОІ is algebraic over F . By brute
force (try ai = 1, aj = 0, j > i for i = 1, 2, . . . , n), we will eventually arrive at the
minimal polynomial.
6. Deп¬Ѓne П• : F (X) в†’ E by П•(f (X)/g(X)) = f (О±)/g(О±). Note that П• is well-deп¬Ѓned,
since if g is a nonzero polynomial, then g(О±) = 0 (because О± is transcendental over F ).
By (3.1.2), П• is a monomorphism. Since П•(F (X)) = F (О±), it follows that F (X) and
F (О±) are isomorphic.
7. The kernel of П• is I, and as in (3.1.3), F [X]/I is a п¬Ѓeld. The image of П• is F [О±], and
by the п¬Ѓrst isomorphism theorem for rings, F [О±] is isomorphic to F [X]/I. Therefore
F [О±] is a п¬Ѓeld, and consequently F [О±] = F (О±).
8. If f = gh, then (g + I)(h + I) = 0 in F [X]/I, so F [X]/I is not a п¬Ѓeld. By (2.4.3), I
is not maximal.
9. The minimal polynomial over F belongs to F [X] вЉ† E[X], and has О± as a root. Thus
min(О±, E) divides min(О±, F ).
10. The result is true for n = 1; see (3.1.7). Let E = F [О±1 , . . . , О±nв€’1 ], so that
[F [О±1 , . . . , О±n ] : F ] = [F [О±1 , . . . , О±n ] : E][E : F ] = [E[О±n ] : E][E : F ]. But [E[О±n ] : E]
is the degree of the minimal polynomial of О±n over E, which is at most the degree of
the minimal polynomial of О±n over F , by Problem 9. An application of the induction
hypothesis completes the proof.

Section 3.2
1. f (X) = (X в€’ 2)2 , so we may take the splitting п¬Ѓeld K to be Q itself.
в€љ в€љ в€љ
2. f (X) = (X в€’ 1)2 + 3, with roots 1 В± i 3, so K = Q(i 3). Now i 3 в€€ Q since /
в€љ2 в€љ
(i 3) = в€’3 < 0, so [K : Q] в‰Ґ 2. But i 3 is a root of X + 3, so [K : Q] в‰¤ 2.
2

Therefore [K : Q] = 2.
3. Let О± be the positive 4th root of 2. The roots of f (X) are О±, iО±, в€’О± and в€’iО±. Thus
K = Q(О±, i). Now f (X) is irreducible by Eisenstein, so [Q(О±) : Q] = 4. Since
i в€€ Q(О±) and i is a root of X 2 + 1 в€€ Q(О±)[X], we have [K : Q(О±)] = 2. By (3.1.9),
/
[K : Q] = 2 Г— 4 = 8.
4. The argument of (3.2.1) may be reproduced, with the polynomial f replaced by the
family C of polynomials, and the roots О±1 , . . . , О±k of f by the collection of all roots of
the polynomials in the family C.
5. Take f = f1 В· В· В· fr . Since О± is a root of f iп¬Ђ О± is a root of some fi , the result follows.
в€љ в€љ в€љ
6. If the degree is less thanв€љ it must be 2 (since m в€€ Q). In this case, n = a + b m,
4, /
so n = a2 + b2 m + 2ab m. Since m is square-free, we must have a = 0 or b = 0,
в€љ в€љ
and the latter is impossible because n is square-free. Thus n = b m, so n = b2 m, a
contradiction of the hypothesis that m and n are distinct and square-free.
15

Section 3.3
1. If О±1 , . . . , О±n form a basis for E over F , then E is generated over F by the О±i . Each
О±i is algebraic over F because F (О±i ) вЉ† E, and (3.1.10) applies.
2. There are only countably many polynomials with rational coeп¬ѓcients, and each such
polynomial has only п¬Ѓnitely many roots. Since an algebraic number must be a root of
one of these polynomials, the set of algebraic numbers is countable. Since the complex
п¬Ѓeld is uncountably inп¬Ѓnite, there are uncountably many transcendental numbers.
3. The complex п¬Ѓeld C is algebraically closed, and C is an extension of the rational п¬Ѓeld Q.
But C is not algebraic over Q, by Problem 2.
4. The algebraic numbers A form a п¬Ѓeld by (3.3.4), and A is algebraic over Q by deп¬Ѓnition.
But it follows from Section 2.9, Problem 2, that A contains subп¬Ѓelds of arbitrarily high
degree (in fact subп¬Ѓelds of every degree) over Q, so that A/Q is not п¬Ѓnite.
5. This can be veriп¬Ѓed by transп¬Ѓnite induction. A splitting п¬Ѓeld is always an algebraic ex-
tension (see (3.2.2)), and the п¬Ѓeld F<f is algebraic over F by the induction hypothesis.
The result follows from (3.3.5).
6. By deп¬Ѓnition of algebraic number, A is an algebraic extension of Q. If О± is algebraic
over A, then as in (3.3.5), О± is algebraic over Q, so О± в€€ A. Thus A has no proper
algebraic extensions, so by (3.3.1), A is algebraically closed.
7. Since E is an extension of F we have |F | в‰¤ |E|. Suppose that О± в€€ E and the minimal
polynomial f of О± has roots О±1 , . . . , О±n , with О± = О±i . Then the map О± в†’ (f, i) is
injective, since f and i determine О±. It follows that |E| в‰¤ |F [X]|в„µ0 = |F [X]|. But
for each n, the set of polynomials of degree n over F has cardinality |F |n+1 = |F |, so
|F [X]| = |F |в„µ0 = |F |. Thus |E| = |F |.
8. Let C be an algebraic closure of F , and let A be the set of roots in C of all polynomials
in S. Then F (A), the п¬Ѓeld generated over F by the elements of A, is a splitting п¬Ѓeld
for S over F ; see Section 3.2, Problem 4.
n
9. If F is a п¬Ѓnite п¬Ѓeld with elements a1 , . . . , an , the polynomial f (X) = 1 + i=1 (X в€’ ai )
has no root in F , so F cannot be algebraically closed.

Section 3.4
1. Let f (X) = (X в€’ 1)p over Fp .
2. О± is a root of X p в€’ О±p = (X в€’ О±)p , so m(X) divides (X в€’ О±)p .
3. By Problem 2, m(X) = (X в€’ О±)r for some r. We are assuming that О± is separable
over F (О±p ), so m(X) must be simply X в€’ О±. But then О± в€€ F (О±p ).
4. The вЂњifвЂќ part follows from the proof of (3.4.5), so assume that F is perfect and let
b в€€ F . Let f (X) = X p в€’ b and adjoin a root О± of f . Then О±p = b, so F (О±p ) =
F (b) = F . By hypothesis, О± is separable over F = F (О±p ), so by Problem 3, О± в€€ F .
But then b is the pth power of an element of F .
5. If О±1 , . . . , О±n is a basis for E over F , then by the binomial expansion mod p,
p p
p
K = F (О±1 , . . . , О±n ). Now since E/F is algebraic, the elements of F (О±1 ) can be ex-
p p
pressed as polynomials in О±1 with coeп¬ѓcients in F . Continuing, О±2 is algebraic over
16

p p p
F , hence over F (О±1 ), so each element of F (О±1 , О±2 ) can be written as a polynomial in
p p
О±2 with coeп¬ѓcients in F (О±1 ). Such an element has the form
pr ps
brs О±1 О±2
s r

with the brs в€€ F . An induction argument completes the proof.
6. Extend the yi to a basis y1 , . . . , yn for E over F . By Problem 5, every element of
p p
E(= F (E p )) has the form y = a1 y1 + В· В· В· + an yn with the ai в€€ F . Thus {y1 , . . . , yn }
p p

spans E over F . It follows that this set contains a basis, hence (since there are exactly
n vectors in the set) the set is a basis for E over F . The result follows.
7. Assume the extension is separable, and let О± в€€ E. Then О± is separable over F , hence
over F (О±p ), so by Problem 3, О± в€€ F (E p ). Thus E = F (E p ). Conversely, suppose that
E = F (E p ) and the element О± в€€ E has an inseparable minimal polynomial m(X).
By (3.4.3), m(X) is of the form b0 + b1 X p + В· В· В· + brв€’1 X (rв€’1)p + X rp . Since m(О±) = 0,
the elements 1, О±p , . . . , О±rp are linearly dependent over F . But by minimality of
m(X), 1, О±, . . . , О±rpв€’1 are linearly independent over F , hence 1, О±, . . . , О±r are linearly
independent over F . (Note that rp в€’ 1 в‰Ґ 2r в€’ 1 в‰Ґ r.) By Problem 6, 1, О±p , . . . , О±rp
are linearly independent over F , which is a contradiction. Thus E/F is separable.
8. We may assume that F has prime characteristic p. By Problem 7, E = K(E p ) and
K = F (K p ). Thus E = F (K p , E p ) = F (E p ) since K в‰¤ E. Again by Problem 7,
E/F is separable.
m
9. If g can be factored, so can f , and therefore g is irreducible. If f (X) = g(X p ) with
m maximal, then g в€€ F [X p ]. By (3.4.3) part (2), g is separable.
/
m
10. Suppose that the roots of g in a splitting п¬Ѓeld are c1 , . . . , cr . Then f (X) = g(X p ) =
m m
(X p в€’ c1 ) В· В· В· (X p в€’ cr ). By separability of g, the cj must be distinct, and since
m
f (О±) = 0, we have О±p = cj for all j. This is impossible unless r = 1, in which case
m m
f (X) = X p в€’ c1 . But f в€€ F [X], so О±p = c1 в€€ F .
n n n
11. If О±p = c в€€ F , then О± is a root of X p в€’ c = (X в€’ О±)p , so min(О±, F ) is a power
of X в€’ О±, and therefore has only one distinct root О±. The converse follows from
Problem 10 with f = min(О±, F ).

Section 3.5
в€љ
1. Take F = Q, K = Q( 3 2) (see (3.5.3)), and let E be any extension of K that is normal
over F , for example, E = C.
2. The polynomial f (X) = X 2 в€’ a is irreducible, else it would factor as (X в€’ b)(X в€’ c)
with b + c = 0, bc = a, i.e., (X в€’ b)(X + b) with b2 = a, contradicting the hypothesis.
в€љ
Thus E is obtained from Q by adjoining a root of f . The other root of f is в€’ a, so
that E is a splitting п¬Ѓeld of f over Q. By (3.5.7), E/Q is normal.
в€љ в€љ
3. Take F = Q, K = Q( 2), E = Q( 4 2). Then K/F is normal by Problem 2, and
E/K is normal by a similar argument. But E/F is not normal, since the two complex
roots of X 4 в€’ 2 do not belong to E. The same argument works with 2 replaced by any
positive integer that is not a perfect square.
17

4. There are at most n embeddings of E in C extending Пѓ. The proof is the same, except
that now g has at most r distinct roots in C, so there are at most r possible choices
of ОІ. The induction hypothesis yields at most n/r extensions from F (О±) to E, and the
result follows.
5. Since the rationals have characteristic zero, the extension is separable. Since E is the
splitting п¬Ѓeld of (X 2 в€’ 2)(X 2 в€’ 3) over Q, the extension is normal, hence Galois.
в€љ в€љ
6. Since 3 в€€ Q( 2), the extension has degree 4. By (3.5.9), there are exactly four
/
Q-automorphisms in the Galois group. By (3.5.1), each such Q-automorphism must
permute the roots of X 2 в€’ 2 and must also permute the roots of X 2 в€’ 3. There are only
four possible ways this can be done. Since a Q-automorphism is completely speciп¬Ѓed
в€љ в€љ
by its action on 2 and 3, the Galois group may be described as follows:
в€љ в€љ в€љ в€љ
(1) 2 в†’ 2, 3 в†’ 3;
в€љ в€љ в€љ в€љ
(2) 2 в†’ 2, 3 в†’ в€’ 3;
в€љ в€љ в€љ в€љ
(3) 2 в†’ в€’ 2, 3 в†’ 3;
в€љ в€љ в€љ в€љ
(4) 2 в†’ в€’ 2, 3 в†’ в€’ 3.
Since the product (composition) of any two of automorphisms (2),(3),(4) is the third,
the Galois group is isomorphic to the four group (Section 1.2, Problem 6).
7. Yes, up to isomorphism. If f is the polynomial given in (3.5.11), any normal closure is
a splitting п¬Ѓeld for f over F , and the result follows from (3.2.5).
8. If f is irreducible over F and has a root in E1 в€© E2 , then f splits over both E1 and E2 ,
hence all roots of f lie in E1 в€© E2 . Thus f splits over E1 в€© E2 , and the result follows.

Section 4.1
1. If x в€€ R, take r(x + I) to be rx + I to produce a left R-module, and (x + I)r = xr + I
for a right R-module. Since I is an ideal, the scalar multiplication is well-deп¬Ѓned, and
the requirements for a module can be veriп¬Ѓed using the basic properties of quotient
rings.
2. If A is an algebra over F , the map x в†’ x1 of F into A is a homomorphism, and since F
is a п¬Ѓeld, it is a monomorphism (see (3.1.2)). Thus A contains a copy of F . Conversely,
if F is a subring of A, then A is a vector space over F , and the compatibility conditions
are automatic since A is commutative.
3. Let R = Z and let M be the additive group of integers mod m, where m is composite,
say m = ab with a, b > 1. Take x = a (mod m) and r = b.
4. Any set containing 0 is linearly dependent, so assume a/b and c/d are nonzero rationals.
Since a/b is rational, the result follows.
c/d
5. In view of Problem 4, the only hope is that a single nonzero rational number a/b spans
M over Z. But this cannot happen, since an integer multiple of a/b must be a fraction
whose denominator is a divisor of b.
6. If a в€€ A вЉ† C and x в€€ B в€© C, then ax в€€ (AB) в€© C. Conversely, let c = ab в€€ (AB) в€© C.
Then b = aв€’1 c в€€ C since A вЉ† C. Thus ab в€€ A(B в€© C).
18

7. If f (X) = a0 + a1 X + В· В· В· + an X n and v в€€ V , take

f (X)v = f (T )v = a0 Iv + a1 T v + В· В· В· + an T n v

where I is the identity transformation and T i is the composition of T with itself i
times.

Section 4.2
1. Let W be a submodule of M/N . By the correspondence theorem, W = L/N for some
submodule L of M with L в‰Ґ N . Since L = L + N , we have W = (L + N )/N .
2. No. If S is any submodule of M , then S + N is a submodule of M containing N ,
so S + N corresponds to W = (S + N )/N . We know that W can also be written as
(L + N )/N where L в‰Ґ N . (For example, L = S + N .) By the correspondence theorem,
S + N = L + N , and there is no contradiction.
3. If A в€€ Mn (R), then AE11 retains column 1 of A, with all other columns zero.
4. To identify the annihilator of E11 , observe that by Problem 4, AE11 = 0 iп¬Ђ column 1
of A is zero. For the annihilator of M , note that Ej1 в€€ M for every j, and AEj1 has
column j of A as column 1, with zeros elsewhere. (See Section 2.2, Problem 4.) Thus
if A annihilates everything in M , then column j of A is zero for every j, so that A is
the zero matrix.
5. R/I is an R-module by Problem 1 of Section 4.1 If r в€€ R then r + I = r(1 + I), so
R/I is cyclic with generator 1 + I.
6. We must show that scalar multiplication is well-deп¬Ѓned, that is, if r в€€ I, then rm = 0
for all m в€€ M . Thus I must annihilate M , in other words, IM = 0, where the
rj mj , rj в€€ R, mj в€€ M .
submodule IM is the set of all п¬Ѓnite sums
7. No, since (r + I)m coincides with rm.

Section 4.3
1. Essentially the same proof as in (4.3.3) works. If z1 + В· В· В· + zn = 0, with zi в€€ Mi , then
zn is a sum of terms from previous modules, and is therefore 0. Inductively, every zi
is 0. (In the terminology of (4.3.3), zi is xi в€’ yi .)
2. Only when A = {0}. If A has n elements, then by LagrangeвЂ™s theorem, nx = 0 for
every x в€€ A, so there are no linearly independent sets (except the empty set).
3. This follows because (в€’s)r + rs = 0.
4. If I is not a principal ideal, then I can never be free. For if I has a basis consisting of
a single element, then I is principal, a contradiction. But by Problem 3, there cannot
be a basis with more than one element. If I = a is principal, then I is free if and
only if a is not a zero-divisor.
5. Z, or any direct sum of copies of Z, is a free Z-module. The additive group of rational
numbers is not a free Z-module, by Problem 5 of Section 4.1
19

6. The вЂњonly ifвЂќ part was done in (4.3.6), so assume that M has the given property. Con-
struct a free module M = вЉ•iв€€S Ri where Ri = R for all i. Then the map f : S в†’ M
with f (i) = ei (where ei has 1 in its ith component and zeros elsewhere) extends to a
homomorphism (also called f ) from M to M . Let g : M в†’ M be the module homo-
morphism determined by g(ei ) = i. Then g в—¦ f is the identity on S, hence on M , by
the uniqueness assumption. Similarly, f в—¦ g = 1.
7. An element of M is speciп¬Ѓed by choosing a п¬Ѓnite subset F of О±, and then selecting an
element bi в€€ R for each i в€€ F . The п¬Ѓrst choice can be made in О± ways, and the second
in |R||F | = |R| ways. Thus |M | = О±|R| = max(О±, |R|).
8. We may take B to the set of вЂњvectorsвЂќ (ei ) with 1 in position i and zeros elsewhere.
Thus there is a basis element for each copy of R, so |B| = О±.

Section 4.4
1. To prove that the condition is necessary, take the determinant of the equation
P P в€’1 = I. Suп¬ѓciency follows from CramerвЂ™s rule.
2. A homomorphism f : V в†’ W is determined by its action on elements of the form
(0, . . . , 0, xj , 0, . . . , 0). Thus we must examine homomorphisms from Vj to вЉ•m Wi .
i=1
Because of the direct sum, such mappings are assembled from homomorphisms from
Vj to Wi , i = 1, . . . , m. Thus f may be identiп¬Ѓed with an mГ—n matrix whose ij element
is a homomorphism from Vj to Wi . Formally, we have an abelian group isomorphism
HomR (V, W ) в€ј [HomR (Vj , Wi )].
=
3. In Problem 2, replace V and W by V n and take all Wi and Vj to be V . This gives an
abelian group isomorphism of the desired form. Now if f corresponds to [fij ] where
fij : Vj в†’ Vi , and g corresponds to [gij ], then the composition g в—¦ f is assembled
from homomorphisms gik в—¦ fkj : Vj в†’ Vk в†’ Vi . Thus composition of homomorphisms
corresponds to multiplication of matrices, and we have a ring isomorphism.
4. In (4.4.1), take n = m = 1 and M = R.
5. Since f (x) = f (x1) = xf (1), we may take r = f (1).
6. This follows from Problems 3 and 4, with V = R.
7. If the endomorphism f is represented by the matrix A and g by B, then for any c в€€ R,
we have c(g в—¦ f ) = (cg) в—¦ f = g в—¦ (cf ), so EndR (M ) is an R-algebra. Furthermore,
cf is represented by cA, so the ring isomorphism is also an R-module homomorphism,
hence an R-algebra isomorphism.

Section 4.5
1. Add column 2 to column 1, then add -3 times column 1 to column 2, then add в€’4
times row 2 to row 3. The Smith normal form is
пЈ® пЈ№
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