NOTRE DAME MATHEMATICAL LECTURES
Number 2
GALOIS THEORY
Lectures delivered at the University of Notre Dame
by
D R . E M I L ARTIN
Professor of Mathematics, Princeton University
Edited and supplemented with a Section on Applications
by
DR. ARTHUR N. MILGRAM
Associate Professor of Mathematics, University of Minnesota
Second Edition
With Additions and Revisions
UNIVERSITY OF NOTRE DAME PRESS
NOTRE DAME LONDON
Copyright 1942, 1944
UNIVERSITY OF NOTRE DAME
Second Printing, February 1964
Third Printing, July 1965
Fourth Printing, August 1966
New composition with corrections
Fifth Printing, March 1970
Sixth Printing, January 197 1
Printed in the United States of America by
NAPCO Graphie Arts, Inc., Milwaukee, Wisconsin
TABLE OF CONTENTS
(The sections marked with an asterisk
have been herein added to the content
of the first edition)
Page
1 LINEAR ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
A . Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1
B. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
C. Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . 2
D. Dependence and Independence of Vectors . . , . . . . . . . . . 4
9
E. Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . .
11
F.* Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FIELD THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . <. . . . . . . . .
II 21
21
A. Extension Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
25
C. Algebraic Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
D. Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
E. Unique Decomposition of Polynomials
33
into Irreducible Factors . . . . . . . . . . . , . . . . . . . . . .
F. Group Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
G.* Applications and Examples to Theorem 13 . . . . . . . . . . . . 38
H. Normal Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
49
Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
. . .., . . 56
J. Roots of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
K. Noether Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
L. Kummer™s Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
M . Simple Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
N. Existence of a Normal Basis . . . . . . . . . . . , . . . . . . . . . . . 66
Q. Theorem on Natural Irrationalities . . . . . . . . . . . . . . . . . . . 67
111 APPLICATIONS
, ........... 69
By A. N. Milgram., . . . . . . . . . . . . . . . . . . . . .
69
A. Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B. Permutation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
C. Solution of Equations by Radicals . . . . . . . . . . . . . . . . . . . 72
D. The General Equation of Degree n. . . . . . . . . . . . . . . . . . . 74
E. Solvable Equations of Prime Degree . . . . . . . . . . . . . . . . . 76
F. Ruler and Compass Construction . . . . . . . . . . . . . . . . . . . . 80
1 LINEAR ALGEBRA
A. Fie™lds
*
A field is a set of elements in which a pair of operations called
multiplication and addition is defined analogous to the operations of
multipl:ication and addition in the real number system (which is itself
an example of a field). In each field F there exist unique elements
called o and 1 which, under the operations of addition and multiplica
tion, behave with respect to a11 the other elements of F exactly as
their correspondents in the real number system. In two respects, the
analogy is not complete: 1) multiplication is not assumed to be commu
tative in every field, and 2) a field may have only a finite number
of elements.
More exactly, a field is a set of elements which, under the above
mentioned operation of addition, forms an additive abelian group and
for which the elements, exclusive of zero, form a multiplicative group
and, finally, in which the two group operations are connected by the
distributive law. Furthermore, the product of o and any element is de
fined to be o.
If multiplication in the field is commutative, then the field is
called a commutative field.
B. Vector Spaces.
If V is an additive abelian group with elements A, B, . . . ,
F a field with elements a, b, . . . , and if for each a c F and A e V
2
the product aA denotes an element of V, then V is called a (left)
vector space over F if the following assumptions hold:
1) a(A + B) = aA + aB
2) (a + b)A = aA + bA
3) a(bA) = (ab)A
4) 1A = A
The reader may readily verify that if V is a vector space over F, then
oA = 0 and a0 = 0 where o is the zero element of F and 0 that of V.
For example, the first relation follows from the equations:
aA = (a + o)A = aA + oA
Sometimes products between elements of F and V are written in
the form Aa in which case V is called a right vector space over F to
distinguish it from the previous case where multiplication by field ele
ments is from the left. If, in the discussion, left and right vector
spaces do not occur simultaneously, we shall simply use the term
“vector space.”
C. Homogeneous Linear Equations.
If in a field F, aij, i = 1,2,. . . , m, j = 1,2, . . . , n are m . n ele
ments, it is frequently necessary to know conditions guaranteeing the
existence of elements in F such that the following equations are satisfied:
a,, xi + a,, x2 + . . . + alnxn = 0.
(1) . *
a ml˜l + amzx2 + . . . + amnxn = 0.
The reader Will recall that such equations are called linear
homogeneous equations, and a set of elements, xi, x2,. . . , xr,
of F, for which a11 the above equations are true, is called
3
a solution of the system. If not a11 of the elements xi, xg, . . . , xn
are o the solution is called nontrivial; otherwise, it is called trivial.
THEOREM 1. A system of linear homogeneous equations always
has a nontrivial solution if the number of unknowns exceeds the num
ber of equations.
The proof of this follows the method familiar to most high school
students, namely, successive elimination of unknowns. If no equations
in n > 0 variables are prescribed, then our unknowns are unrestricted
and we may set them a11 = 1.
We shall proceed by complete induction. Let us suppose that
each system of k equations in more than k unknowns has a nontrivial
solution when k < m. In the system of equations (1) we assume that
n > m, and denote the expression a,ixi + . . . + ainxn by L,, i = 1,2,. . .,m.
We seek elements xi, . . . , x,, not a11 o such that L, = L, = . . . = Lm = o.
If aij = o for each i and j, then any choice of xi , . . . , xr, Will serve as
a solution. If not a11 aij are o, then we may assume that ail f o, for
the order in which the equations are written or in which the unknowns
are numbered has no influence on the existence or nonexistence of a
simultaneous solution. We cari find a nontrivial solution to our given
system of equations, if and only if we cari find a nontrivial solution
to the following system:
L, = 0
L,  a,,a,;lL, = 0
. . . . .
Lm  amia,;lL, =0
For, if xi,. . . , x,, is a solution of these latter equations then, since
L, = o, the second term in each of the remaining equations is o and,
hence, L, = L, = . . . = Lm = o. Conversely, if (1) is satisfied, then
the new system is clearly satisfied. The reader Will notice that the
new system was set up in such a way as to “eliminate” x1 from the
last ml equations. Furthermore, if a nontrivial solution of the last
ml equations, when viewed as equations in x2, . . . , xn, exists then
taking xi =  ai;˜( ai2xz + ar3x3 + . . . + alnxn) would give us a
solution to the whole system. However, the last ml equations have
a solution by our inductive assumption, from which the theorem follows.
Remark: If the linear homogeneous equations had been written
in the form xxjaij = o, j = 1,2, . . . , n, the above theorem would still
hold and with the same proof although with the order in which terms
are written changed in a few instances.
D. Dependence and Independence of Vectors.
In a vector space V over a field F, the vectors A,, . . . , An are
called dependent if there exist elements xi, . . . , x”, not a11 o, of F such
that xiA, + x2A, + . . . + xnAn = 0. If the vectors A,, . . . ,An are
not dependent, they are called independent.
The dimension of a vector space V over a field F is the maximum
number of independent elements in V. Thus, the dimension of V is n if
there are n independent elements in V, but no set of more than n
independent elements.
A system A,, . . . , A, of elements in V is called a
generating system of V if each element A of V cari be expressed
5
linearly in terms of A,, . . . , Am, i.e., A = Ca.A. for a suitable choice
i=ll 1
ofa,, i = l , . . . , m , i n F .
THEOREM 2. In any generating system the maximum number of
independent vectors is equal to the dimension of the vector space.
Let A,, . . . , A,,, be a generating system of a vector space V of
dimension n. Let r be the maximum number of independent elements in
the generating system. By a suitable reordering of the generators we may as
sumek,, . . . , Ar independent. By the definition of dimension it follows that
r < n. For each j, A,, . . . , A,. A,+j are dependent, and in the relation

a,A, + a,A, + ..* + arAr + a,+j A,+j = 0
expressing this, a ,+j # o, for the contrary would assert the dependence
of A,, . . . ,Ar. Thus,
A,+j =  ar+y[a,A, + a,A, + . . . + arAr].
It follows that A,, . . . , Ar is also a generating system since in the
linear relation for any element of V the terms involving Ar+j, j f o, cari
a11 be replaced by linear expressions in A,, . . . , Ar.
Now, let B,, . . . , B, be any system of vectors in V where t > r,
then there exist aij such that Bj =iglaij Ai, j = 1,2, . . . , t, since the
Ai™ s form a generating system. If we cari show that B,, . . . , B, are
dependent, this Will give us r  n, and the theorem Will follow from
>
this together with the previous inequality r < n. Thus, we must ex
hibit the existence of a nontrivial solution out of F of
the equation
xiB, + x2B, + . . . + xrB, = 0.
6
TO this end, it Will be sufficient to choose the xi™s as to satisfy
SO
the linear equationsiir xj aij = o, i = 1,2,. . . , r, since these ex
pressions Will be the coefficients of Ai when in E x. B. the Bj ˜s are
r j=l J J
replaced by 2 aij Ai and terms are collected. A solution to the equa
i=l
tions 2 xjaij = 0, i = 1,2,. . . , r, always exists by Theorem 1.
.i=l
Remark: Any n independent vectors A,, . . . , A,, in an n dimen
sional vector space form a generating system. For any vector A, the
vectors A, A,, . . . , A,, are dependent and the coefficient of A, in the
dependence relation, cannot be zero. Solving for A in terms of
A l>“˜> A,, exhibits A,, . . . ,An as a generating system.
A subset of a vector space is called a subspace if it is a sub
group of the vector space and if, in addition, the multiplication of any
element in the subset by any element of the field is also in the subset.
I f A i , . . . , AS are elements of a vector space V, then the set of a11 ele
ments of the form a, A, + . . . + asAS clearly forms a subspace of V.
It is also evident, from the definition of dimension, that the dimension
of any subspace never exceeds the dimension of the whole
vector space.
An stuple of elements ( a,, . . . , as ) in a field F Will be called
a row vector. The totality of such stuples form a vector space if

we define
a) (a,,a, ,..., as) = (b,,b, ,..., bS)ifandonlyif
a, = b,, i = 1,. . . , s,
B> (alta2,...,as) + (bl,b2,...,bs) = (a1 + b,,a, + b,,
. . ..aS + bs),
y) b(a,,a, ,..., as) = (ba,,ba, ,..., baS),forban
element of F.
When the stuples are written vertically,
they Will be called column vectors.
THEOREM 3. The row (column) vector space F” of a11 ntuples
from a field F is a vector space of dimension n over F.
The n elements
Cl = (l,o,o ,...> 0)
E2 = (o,l,o >..., 0)
6, = (o,o,...,o,l)
are independent and generate F”. Both remarks follow from the relation
(a1,a2,. . . ,an) = Xaici.
We cal1 a rectangular array
of elements of a field F a matrix. By the right row rank of a matrix, we
mean the maximum number of independent row vectors among the rows
(ail,..., a,,) of the matrix when multiplication by field elements is
from the right. Similarly, we define left row rank, right column rank and
left column rank.
THEOREM 4. In any matrix the right column rank equals the left
row rank and the left column rank equals the right row rank. If the field
8
is commutative, these four numbers are equal to each other and are
called the rank of the matrix.
Cal1 the column vectors of the matrix C,, . ˜. , Cn and the row
vectors R,, . . . , Rm. The column vector 0 is o and any
0
(:)
0
dependence Crx, + C,x, + . . . + Cnx, = 0 is equivalent to a
solution of the equations
arrxr + a12x2 + . . . + a lnxn = O
(1) :
0.
amlxl + amZx2 + . . . + a nmxn =
Any change in the order in which the rows of the matrix are written
gives rise to the same system of equations and, hence, does not change
the column rank of the matrix, but also does not change the row rank
since the changed matrix would have the same set of row vectors. Cal1
c the right column rank and r the left row rank of the matrix. By the
above remarks we may assume that the first r rows are independent row
vectors. The row vector space generated by a11 the rows of the matrix
has, by Theorem 1, the dimension r and is even generated by the first
r rows. Thus, each row after the rth .1s linearly expressible in terms of
the first r rows. Consequently, any solution of the first r equations in
(1) Will be a solution of the entire system since any of the last nr
equations is obtainable as a linear combination of the first r. Con
versely, any solution of (1) Will also be a solution of the first r
equations. This means that the matrix
9
(4
%la12 * .%n
. .
. .
. .
arr ar2. . . a rn
consisting of the first r rows of the original matrix has the same right
column rank as the original. It has also the same left row rank since
the r rows were chosen independent. But the column rank of the ampu
tated matrix carmot exceed r by Theorem 3. Hence, c < r. Similarly,

calling c™ the left column rank and r™ the right row rank, c™ < r™.
If we form the transpose of the original matrix, that is, replace rows by
columns and columns by rows, then the left row rank of the transposed
matrix equals the left column rank of the original. If then to the
transposed matrix we apply the above considerations we arrive at
r  c and r™ < c™.
< .
E. 
Nonhomogeneous Linear Equations.
The system of nonhomogeneous linear equations
arrxi + ar2x2 + . . . + alnxn = bl
azlxl + . . . . . . . . . . . + aznxn = b2
(2) :
amlxl + . . . . . . . . . . . + a mm=
x iln8
has a solution if and only if the column vector lies
in the space generated by the vectors
(..) (i:)
10
This means that there is a solution if and only if the right column rank of
the matrix is the same as the
51.. .%n
a
ii ml™ . . amn
4
right column rank of the augmented matrix 51. . * %A
. .
. .
. .
a
i! ml* * ™ arnnb, i
since the vector space generated by the original must be the same as
the vector space generated by the augmented matrix and in either case
the dimension is the same as the rank of the matrix by Theorem 2.
By Theorem 4, this means that the row tanks are equal. Con
versely, if the row rank of the augmented matrix is the same as the row
rank of the original matrix, the column ranks Will be the same and the
equations Will have a solution.
If the equations (2) have a solution, then any relation among the
rows of the original matrix subsists among the rows of the augmented
matrix. For equations (2) this merely means that like combinations
of equals are equal. Conversely, if each relation which subsists be
tween the rows of the original matrix also subsists between the rows
of the augmented matrix, then the row rank of the augmented matrix
is the same as the row rank of the original matrix. In terms of the
equations this means that there Will exist a solution if and only if
the equations are consistent, Le., if and only if any dependence
between the left hand sides of the equations also holds between the
right sides.
11
THEOREM 5. If in equations (2) m = n, there exists a unique
solution if and only if the corresponding homogeneous equations
arrxr + arzxz + . . . + alnxn = 0
anlxl + an2x2 + . . . + annxn = 0
have only the trivial solution.
If they have only the trivial solution, then the column vectors
are independent. It follows that the original n equations in n unknowns
Will have a unique solution if they have any solution, since the differ
ence, term by term, of two distinct solutions would be a nontrivial
solution of the homogeneous equations. A solution would exist since
the n independent column vectors form a generating system for the
ndimensional space of column vectors.
Conversely, let us suppose our equations have one and only one
solution. In this case, the homogeneous equations added term by
term to a solution of the original equations would yield a new solu
tion to the original equations. Hence, the homogeneous equations have
only the trivial solution.
F. Qeterminants. l)
The theory of determinants that we shall develop in this chapter
is not needed in Galois theory. The reader may, therefore, omit this
section if he desires.
SO
We assume our field to be c o m m ut a t i v e and consider the
square matrix
1) Of the preceding theory only Theorem 1, for
homogeneous equations and the notion of
linear dependence are assumed known.
12
(J
allalz* . . . %n
a21a22˜**.%
(1)
............
an1 an2. . . . a nn
of n rows and n columns. We shall define a certain function of this
matrix whose value is an element of our field. The function Will be
called the determinant and Will be denoted by
%1%2. . . .%n
%!1a22.˜a2n
............
n,?l,* *. ˜?ln
An) if we wish to consider it as a function of the
o r b y D(A,,A,,...
column vectors A,, A,, . . . A,, of (1). If we keep a11 the columns but A,
constant and consider the determinant as a function of A,, then we
Write DJ Ak) and sometimes even only D.
Definition. A function of the column vectors is a determinant if
it satisfies the following three axioms:
1. Viewed as a function of any column A, it is linear and homogeneous, i.e..,
( 3 ) &(A, + 4) = Dk(Ak) + &(A;)
(4) D,(cA,) = cD,(A, >
2. Its value is = 01) if the adjacent columns A, and Ak+l are equal.
3. Its value is = 1 if a11 A, are the unit vectors U, where
1) H e n c e f o r t h , 0 Will denote t h e zero e l e m e n t
of a field.
(5) “, f);“*=(I) . . . . . ;iii
The question as to whether determinants exist Will be left open
for the present. But we derive consequences from the axioms:
a) If we put c = 0 in (4) we get: a determinant is 0 if one of
the columns is 0.
CA,,,) or a determinant remains unchanged
b) Dk(Ak) = &(A, + 
if we add a multiple of one column to an adjacent column. Indeed
%(A, + CA,,,) = Dk(Ak) + cD,(A,+,) = Dk(Ak)

because of axiom 2.
c) Consider the two columns A, and Ak+i. We may replace them by
A, and Ak+i + A k; subtracting the second from the first we may replace
them by  Ak+i and Ak+i + A,,. adding the first to the second we now
have  Ak+r and A,,. finally, we factor out 1. We conclude: a determi
nant changes sign if we interchange two adjacent columns.
d) A determinant vanishes if any two of its columns are equal.
Indeed, we may bring the two columns side by side after an interchange
of adjacent columns and then use axiom 2. In the same way as in b)
and c) we may now prove the more general rules:
e) Adding a multiple of one column to another does not change
the value of the determinant.
f) Interchanging any two columns changes the sign of D.
14
vn) be a permutation of the subscripts
g> Let(v,,v,,...
(1,2,, . . . n). If we rearrange the columns in D( At,,i, AV2, . . . , A,, )
n
until they are back in the natural order, we see that
WAvl,Av >...y A, ) = +D(A,,A, ,...> An).
2
Here 2 is a definite sign that do:s not depend on the special values
of the A,. If we substitute U, for A, we see that
WJl,JJv2,..., U, ) = i 1 and that the sign depends only on the
1
permutation of the tnit vectors.
Now we replace each vector A, by the following linear combina
tionAkofAr,A2,...,A,:
(6) A; = brkA, + b,,A, + . . . + b,,A,.
In computing D(Ai ,A;, . . . , AA) we first apply axiom 1 on A;
breaking up the determinant into a sum; then in each term we do the
same with A; and on. We get
SO
2
( 7 ) D(A;,A;,...,A;)= D(b, ,A, ,bv22Av >. . . Jj, ,AI, n >
n
v1
,v2*. ,v, 1 1 2
= c b, ;b, 2.. . ..b,nD(Ar, ,A,, . . . ,A, )
I4
1 2 n 1 2
VI,V2, **. ,vn
where each vi runs independently from 1 to n. Should two of the indices
vi be equal, then D( Avl, A, , . . . , AV,) = 0; we need therefore keep
2
only those terras in which ( vi, v2, . . . , is a permutation of
vn)
n). This gives
(1,2,...,
D(A;,A;,...,A;)
(8)
+ bv1,.bv2, . . . . .b, n
2
= D(A,,A2 ,...> A,). n
( vl, ***9Vn)
v,) runs through a11 the permutations of
where(v1,v2,...,
n) and where L stands for the sign associated with that
(1,2,...,
permutation. It is important to remark that we would have arrived at
the same formula (8) if our function D satisfied only the first two
1.5
of our axioms.
Many conclusions may be derived from (8).
W™e first assume axiom 3 and specialize the 4, to the unit vec
tors Uk of (5). This makes 4 = B, where B, is the column vector of
the matrix of the bik. (8) yields now:
2 + bVll+V2®bv n
(9) WBl,B2,...,B,,)=(V n
1 >UZ™...™ vn > 
giving us an explicit formula for determinants and showing that they are
uniquely determined by our axioms provided they exist at all.
W™ith expression (9) we retum to formula (8) and get
(10) D(A;,A; ,...> A;) = D(A,,A, ,...> A,)D(B,,B, ,...> Bn).
This is the socalled multiplication theorem for determinants. At
the left of (10) we have the determinant of an nrowed matrix whose ele
ments cik are given by
n
Cik =
(11) x ?vbw
v=1
cik is obtained by multiplying the elements of the i  th row of
AJbythoseofthekthcolumnofD(B,,B,,...,B,.,)
WA,,A,,...,
and adding.
Let us now replace D in (8) by a function F(A,, . . . , A,) that
satisfies only the first two axioms. Comparing with (9) we find
F(A;,A; ,..., A;)=F(A, >..., AJD(BI,B2 ,...> B,).
Specializing A, to the unit vectors U, leads to
(12) F(B,,B,, . . . ,B,) = c.D(B,,B,, . . . ,B,)
with c = F(U,,U,,. . . ,U”).
16
Next we specialize (10) in the following way: If i is a certain
subscript from 1 to nl we put A, = U, for k f i, i+ 1
Ai = Ui + Ui+r , Ai+, = 0. Then D( A,, A,, +. . , A, ) = 0 since one col
umn is Q, Thus, D(Ai ,A;, , . . , An) = 0; but this determinant differs
from that of the elements bj, only in the respect that the i+lst row
has been made equal to the itb. We therefore see:
A determinant vanishes if two adjacent rows are equal.
I&ch term in (9) is a product where precisely one factor cornes
from a given row, say, the ith. This shows that the determinant is
linear and homogeneous if §onsidered as function of this row. If,
finally, we Select for eaeh raw the corresponding unit vector, the de
terminant is = 1 since the matrix is the same as that in which the col
umns are unit vectors. This shows that a determinant satisfies our
three axioms if we consider it as function of the row vectors. In view
of the uniqueness it follows:
A determinant remains unchanged if we transpose the row vec
tors into column vectors, that is, if we rotate the matrix about its
main diagonal.
A determinant vanishes if any two rows are equal. It changes
sign if we interchange any two rows. It remains unchanged if we add
a multiple of one row to another.
We shall now prove the existence of determinants. For a 1rowed
matrix a 1 1 the element ai 1 itself is the determinant. Let us assume the
existence of (n  1)  rowed determinants. If we consider the nrowed
matrix (1) we may associate with it certain (n  1)  rowed determinants
in the following way: Let ai, be a particular element in (1). We
17
cancel the ith row and kth column in (1) and take the determinant
of the remaining (n  1)  rowed matrix. This determinant multiplied by
(l)i+k Will be called the cofactor of a ik and be denoted by Ai,.
The distribution of the sign ( 1) i+k follows the chessboard pattern,
namely,
........
Let i be any number from 1 to n. We consider the following
function D of the matrix (1):
D = ailAi, + ai2Ai, + . . + ainAi,.
(13)
[t is the sum of the products of the ith Tow and their cofactors.
Consider this D in its dependence on a given column, say, A,.
For v f k, Au, depends linearly on A, and ai, does not depend on it;
for v =: k, Ai, does not depend on A, but aik is one element of this
column. Thus, axiom 1 is satisfied. Assume next that two adjacent
columns A, and Ak+l are equal. For v f k, k + 1 we have then two
equal columns in Ai, that A,, = 0. The determinants used in the
SO
computation of Ai k and Ai k+l are the same but the signs are opposite
hence, Ai k = Ai k+l whereas ai k = a, k+l™ Thus D = 0 and axiom 2
holds. For the special case A, = U,( v = 1,2,. . , n) we have
= 0 for v f i while a,, = 1, Aii = 1. Hence, D = 1 and
aiV
this is axiom 3. This proves both the existence of an nrowed
18
determinant as well as the truth of formula (13), the socalled develop
ment of a determinant according to its ith row. (13) may be generalized
as follows: In our determinant replace the ith row by the jth row and
develop according to this new row. For i f j that determinant is 0 and
for i = j it is D:
D for j = i
+ ajzAi2 t . . . + ainAi,, =
(14) ajl *il
0 forj f i
If we interchange the rows and the columns we get the
following formula:
D for h = k
(15) a,,* Ik t aZr,A,, + . . + a,hAnk =
0 for h f k
Now let A represent an nrowed and B an mrowed square matrix.
By ( A 1, ( B \ we mean their determinants. Let C be a matrix of n rows
and m columns and form the square matrix of n + m rows
where 0 stands for a zero matrix with m rows and n columns. If we con
sider the determinant of the matrix (16) as a function ofthecolumns of A
only, it satisfies obviously the first two of our axioms. Because of (12)
its value is c . 1A 1where c is the determinant of (16) after substituting
unit vectors for the columns of A. This c still depends on B and con
sidered as function of the rows of B satisfies the first two axioms.
Therefore the determinant of (16) is d. 1A 1. 1B 1where d is the special
case of the determinant of (16) with unit vectors for the columns of A
as well as of B. Subtracting multiples of the columns of A from
C we cari replace C by 0. This shows d = 1 and hence the formula
19
(17)
In a similar fashion we could have shown
A0
(18)
= \A\. JBI.
IC B I
The formulas (17), (18) are special cases of a general theorem
by Lagrange that cari be derived from them. We refer the reader to any
textbook on determinants since in most applications (17) and (18)
are sufficient.
We now investigate what it means for a matrix if its determinant
is zero. We cari easily establish the following facts:
a) If A,, A,, . , . , An are linearly dependent, then
DCA,, A,, . . . t A,) = 0. Indeed one of the vectors, say A,, is then a
linear combination of the other columns; subtracting this linear com
bination from the column A, reduces it to 0 and D = 0.
SO
b) If any vector B cari be expressed as linear combination of
A,, A,, . . . >A, then D(A,,A,,. . ., A,,) # 0. Returning to (6) and
(10) we may Select the values for bi, in such a fashion that every
A! ,= I.Ji. For this choice the left side in (10) is 1 and hence
DCA,,&..., A,) on the right side f 0.
c) Let A,, A,, . . . , A,, be linearly independent and B any other
vector. If we go back to the components in the equation
Aix, + A,x, + . . . + A,.,x,,+ By = 0 we obtain n linear homogeneous
equations in the n + 1 unknowns x i, x 2,. . . , xn, y. Consequently,
there is a nontrivial solution. y must be f 0 or else the
ApAz,...,& would be linearly dependent. But then we cari compute
B out of this equation as a linear combination of A,, A,, . . . , An.
20
Combining these results we obtain:
A determinant vanishes if and only if the column vectors (or the
row vectors) are linearly dependent.
Another way of expressing this result is:
The set of n linear homogeneous equations
( i = 1,2,...,n)
+ ai2x2 + . . . + ainx n = 0
ail 3
in n unknowns has a nontrivial solution if and only if the determinant
of the coefficients is zero.
Another result that cari be deduced is:
If A1,A2>..., A,, are given, then their linear combinations cari
represent any other vector B if and only if D (A *, A,, . . . , An) f 0.
Or:
The set of linear equations
( i = 1,2,...,n)
(19) aiixI + ai2x2 + . . . + ainxn = bi
has a solution for arbitrary values of the bi if and only if the determi
nant ˜of aik is f 0. In that case the solution is unique.
We finally express the solution of (19) by means of determinants
if the determinant D of the aik is f 0.
We multiply for a given k the ith equation with Ai, and add the
equations. (15) gives
( k = 1,2,...,n)
( 2 0 ) D. xk = A,,b, + A,,bz + + Ankb,
and this gives xk. The right side in (12) may also be written as the
determinant obtained from D by replacing the kth column by
b,, b,, . . , b”. The rule thus obtained is known as Cramer™s rule.
21
II FIELD THEORY
A. 
Extension Fields.
If E is a field and F a subset of E which, under the operations
of addition and multiplication in E, itself forms a field, that is, if F is
a subfield of E, then we shall cal1 E an extension of F. The relation
of being an extension of F Will be briefly designated by F C E. If
a, P, y, . . . are elements of E, then by F(a, B, y, . . . ) we shall mean
the set of elements in E which cari be expressed as quotients of poly
nomials in a, p, y, . . with coefficients in F. It is clear that
. . ) is a field and is the smallest extension of F which con
F(a,/3,y,.
tains the elements a, p, y,. . We shall cal1 F(a, 6, y,. . . ) the field
obtained after the adjunction of the elements a, @, y, . . . to F, or the
field generated out of F by the elements a, B, y, . . . . In the sequel a11
fields Will be assumed commutative.
. . ˜.
If F C E, then ignoring the operation of multiplication defined
between the elements of E, we may consider E as a vector space over
F. By the degree of E over F, written (E/F), we shall mean the dimen
sion of the vector space E over F. If (E/F) is finite, E Will be called
a finite extension.
THEOREM 6. If F, B, E are three fields such that
F C ES E, then
C
WF) = (B/F) (E/B) .
Let A1,A2,..., A, be elements of E which are linearly
independent with respect to B and let C 1, C,, . . . , C s be elements
22
of B which are independent with respect to F. Then the products Ci Ai
where i = 1,2, . . . , s and j = 1,2, . . . , r are elements of E which are
independent with respect to F. For if 2 arj C,A, = 0, then
Lj
C( iajj Ci ) Aj is a linear combination of the A, with coefficients in B
j
and because the Aj were independent with respect to B we have
pij Ci = 0 for each j. The independence of the Ci with respect to F
then requires that each aij = 0. Since there are r . s elements C,A, we
have shown that for each r 5 (E/B) and s 5 (B/F) the degree ( E/F )
> r . s. Therefore, ( E/F) > (B/F) ( E/B). If one of the latter numbers
 
is infinite, the theorem follows. If both (E/B) and (B/F) are finite,
say r and s respectively, we may suppose that the Aj and the Ci are
generating systems of E and B respectively, and we show that the set
of products Ci Aj is a generating system of E over F. Each A E E cari
be expressed linearly in terms of the Aj with coefficients in B. Thus,
A = CBj Aj . Moreover, each Bj being an element of B cari be ex
pressed linearly with coefficients in F in terms of the Ci, i.e.,
Bj = Caij Ci, j = 1,2, . . . , r. Thus, A = Xaij CiAj and the Cil form
an independent generating system of E over F.
Corollary. If F C Fi C F, C . . . C F,, then

(Fn/F) =y (F,/F).(F,/F, > . . . (F,,/F,,i).
B. Polvnomials.
An expression of the form aOxn + a ix”i+ . . . + an is called a
˜ in F of degree n if the coefficients
polynomial
23
a 01. . . > a,., are elements of the field F and ao f 0. Multiplication and
addition of polynomials are performed in the usual way ˜).
˜4 polynomial in F is called reducible in F if it is equal to the
product of two polynomials in F each of degree at least one. Polyno
mials which are not reducible in F are called irreducible in F.
If f (x ) = g(x) . h (x ) is a relation which holds between the
polynomials f (x ), g (x ), h (x ) in a field F, then we shall say that
g (x ) divides f (x ) in F, or that g ( x ) is a factor of f ( x ). It is readily

seen that the degree of f(x) is equal to the sum of the degrees of
g (x ) and h (x ), SO that if neither g ( x ) nor h ( x ) is a constant then
each has a degree less than f(x). It follows from this that by a finite
number of factorizations a polynomial cari always be expressed as a
product of irreducible polynomials in a field F.
For any two polynomials f (x ) and g (x ) the division algorithm
holds, i.e., f(x) = q(x).g(x) + r(x) where q(x) and r(x) are
unique polynomials in F and the degree of r (x ) is less than that of
g(x). ˜This may be shown by the same argument as the reader met in
elementary algebra in the case of the field of real or complex numbers.
We also see that r(x) is the uniquely determined polynomial of a de
gree less than that of g (x ) such that f(x)  r (x ) is divisible by
g (x ). We shall cal1 r (x ) the remainder of f (x ).
1 ) I f we speak o f t h e s e t o f a11 p o l y n o m i a l s
o f d e g r e e lower than II, we s h a l l agree t o
include t h e p o l y n o m i a l 0 i n t h i s s e t ,
t h o u g h i t h a s n o d e g r e e i n t h e proper sense.
24
Also, in the usual way, it may be shown that if a is a root of
the polynomial f (x ) in F than x  u is a factor of f (x ), and as a con
sequence of this that a polynomial in a field cannot have more roots
in the field than its degree.
Lemma. If f(x) is an irreducible polynomial of degree n in F,
 
then there do not exist two polynomials each of degree less than n in
 
F whose product is divisible by f(x).

Let us suppose to the contrary that g(x) and h(x) are poly
nomials of degree less than n whose product is divisible by f(x).
Among a11 polynomials occurring in such pairs we may suppose g(x)
has the smallest degree. Then since f(x) is a factor of g(x) . h (x )
there is a polynomial k(x) such that
k(x).f(x) = g(x).h(x)
By the division algorithm,
f(x) = q(x).g(x) + r(x)
where the degree of r (x ) is less than that of g(x) and r (x ) f 0
since f(x) was assumed irreducible. Multiplying
f(x) = q(x).g(x) + r(x)
by h (x ) and transposing, we have
r(x),h(x) = f(x).h(x)q(x).g(x).h(x)=f(x).h(x)q(x).k(x).f(x)
from which it follows that r(x) . h (x ) is divisible by f (x ). Since r (x )
has a smaller degree than g(x), this last is in contradiction to the
choice of g (x ), from which the lemma follows.
As we saw, many of the theorems of elementary algebra
hold in any field F. However, the socalled Fundamental
Theorem of Algebra, at least in its customary form, does not
hold. It Will be replaced by a theorem due to Kronecker
25
which guarantees for a given polynomial in F the existence of an ex
tension field in which the polynomial has a root. We shall also show
that, in a given field, a polynomial cari net only be factored into irre
ducible factors, but that this factorization is unique up to a constant
factor. The uniqueness depends on the theorem of Kronecker.
C. Algebraic Elements.
Let F be a field and E an extension field of F. If a is an ele
ment of E we may ask whether there are polynomials with coefficients
in F which have a as root. a ia §alled algebraic with respect to F if
.
tkere are such polynomials. New let a be algebraic and Select among ail
polynomials in F which have a as root one, f(x), of lowest degree.
We may assume that the highest coefficient of f(x) La 1. We con
tend that this f(x) ia uniquely determined, that it ts trreducible and
that each polynomial in F w#r the root o is divisible by f (x ). If, in
deed, g ix ) !w a palynomial in F with g(a) = 0, we may divide
g(x) == f(x)q(x) t r(x) where r(x) bas a degree smaller t h a n t h a t
of f(x). Substituting x = a we get r(o) = Q: Dow r(x) has to he
identically 0 since otherwise r (x > would havg the root a apd be of
lower degree thap f (x ): SO g ( x ) ia divisible by f (x )! Thia also shows
the uniqueness of f (x ). If f (x ) were not irreducible, one of the factors
wopld have to vanish for x = a contradicting again the choice of f ( y ).
We consider now the subset E0 of the following elements
8 of E:
26
8 = g(a) = CO + cla + c2a2 + . . . + CnTlanel
where g(x) is a polynomial in F of degree less than n (n being the de
gree of f(x)). This set l$, is closed under addition and multiplication.
The latter may be verified as follows:
If g (x ) and h (x ) are two polynomials of degree less than n we
put g(x)h(x) = q(x)f(x) + r(x) and hence g(a)h(a) = r(a).
Finally we see that the constants cO, c 1, . . , cr,i are uniquely deter
mined by the element 8. Indeed two expressions for the same 0 would
lead after subtracting to an equation for a of lower degree than n.
We remark that the interna1 structure of the set EO does not de
pend on the nature of a but only on the irreducible f (x ). The knowledge
of this polynomial enables us to perform the operations of addition and
multiplication in our set EO. We shall see very soon that E, is a field;
in fact, EO is nothing but the field F(a). As soon as this is shown we
have at once the degree, ( F (a) /F), determined as n, since the space
F(a) is generated by the linearly independent 1, a, a2, . . . , anl.
We shall now try to imitate the set EO without having an exten
sion field E and an element a at our disposal. We shall assume only
an irreducible polynomial
f(x) = x” + a nl xni + . . . + aO
as given.
We Select a symbol 6 and let E, be the set of a11 forma1
polynomials
g(5˜) = CO + c,c + . . + cnJyl
of a degree lower than n. This set forms a group under
addition. We now introduce besides the ordinary multiplication
27
a new kind of multiplication of two elements g (5) and h (4) of E i
by g ([) x h (5). It is defined as the remainder r (6) of the
denoted
g (6) h(c) un d er d ivision by f (4˜ ). We first remark
ordinary product
that any product of m terms gi( c), gz( t), . . . , g,( 0 is again the re
mainder of the ordinary product g i( 5) g,( 5). . . g,( 5). This is true by
definition for m = 2 and follows for every m by induction if we just
prove the easy lemma: The remainder of the product of two remainders
(of two polynomials) is the remainder of the product of these two
polynomials. This fact shows that our new product is associative and
commutative and also that the new product g i( 4) x g,( 4) x . . . x g I[)
Will coincide with the old product g i( 5) g,( 6). . . g,( 6) if the latter
does not exceed n in degree. The distributive law for our multiplication
is readily verified.
The set E i contains our field F and our multiplication in E, has
for F the meaning of the old multiplication. One of the polynomials of
E, is §:. Multiplying it itimes with itself, clearly Will just lead to ti
as long, as i < n. For i = n this is not any more the case since it
leads to the remainder of the polynomial 5”.
This remainder is
5”  f(t) =  a,&“˜ anJn* . . .  a,.
We now give up our old multiplication altogether and keep only
the new one; we also change notation, using the point (or juxtaposition)
as symbol for the new multiplication.
Computing in this sense
c, + Cl[ + c*p + . . . + c,lpl
Will readily lead to this element, since a11 the degrees
28
involved are below n. But
5” =  anyl[nl a,2[n2 . . .  a0.
Transposing we see that f(§) = 0.
We thus have constructed a set E, and an addition and multipli
cation in E r that already satisfies most of the field axioms. E r contains
F as subfield and 5˜ satisfies the equation f (5) = 0. We next have to
show: If g ( 6) $ 0 and h ( .$) are given elements of E r, there is
an element
X(l$> = x, + x1( + . . . + X,J1
in E, such that
g(Ç) *X(t) = h(t).
TO prove it we consider the coefficients xi of X (6) as unknowns and
compute nevertheless the product on the left side, always reducing
higher powers of [ to lower ones. The result is an expression
L, + LJ + . . + L,, (““where each Li is a linear combination of
of the xi with coefficients in F. This expression is to be equal to
h(t); this leads to the n equations with n unknowns:
L, = b,, L, = b,, . . . > L,, = b,,
where the bi are the coefficients of h(E). This system Will be soluble
if the corresponding homogeneous equations
L, = 0, L, = 0, *. . > L,r = 0
bave only the trivial solution.
The homogeneous problem would occur if we should ask for
the set of elements X(Q) satisfying g (5) . X ( 6) = 0. Going back
for a moment to the old multiplication this would mean that the
ordinary product g( 6) X (6) has the remainder 0, and is
29
therefore divisible by f(t). According to the lemma, page 24, this is
only possible for X (6) = 0.
Therefore E, is a field.
Assume now that we have also our old extension E with a root
a of f(x), leading to the set E,. We see that E, has in a certain sense
the same structure as E 1 if we map the element g (6) of E 1 onto the
element g(a) of EO. This mapping Will have the property that the image
of a sum of elements is the sum of the images, and the image of a
product is the product of the images.
Let us therefore define: A mapping u of one field onto another
which is one to one in both directions such that
o(a+˜) = o(a) + CT(˜) and O(U.@) = o(a). o(p) is called an
˜ If the fields in question are not distinct  i.e., are both
isomorphism.
the same field  the isomorphism is called an automorphism. Two
fields for which there exists an isomorphism mapping one on another
are called isomorphic. If not every element of the image field is the image
under o of an element in the first field, then 0 is called an isomorphism
of the first field into the second. Under each isomorphism it is clear
that o(O) = 0 and o( 1) = 1.
We see that E, is also a field and that it is isomorphic to E,.
We now mention a few theorems that follow from our discussion:
THEOREM 7. (Kronecker). If f (x ) is a polynomial in a field F,
there exists an extension E of F in which f(x) has a root.
30
Proof: Construct an extension field in which an irreducible
factor of f ( x ) has a root.
THEOREM 8. Let o be an isomorphism mapping a field F on a
f i e l d F™ Let f (x ) be an irreducible polynomial in F and f ™ (x ) the cor
˜˜
responding polynomial in F ™ . If E = F (B) and E ™ = F ™ (@˜) are exten
˜
sions of F and F™ , respectively, where f(p) = 0 in E and f ™ ( p ˜) = 0 in E™ ,
˜˜
then o™ cari be extended to an isomorphism between E and E ™ .
Proof: E and E™ are both isomorphic to EO.
D. Splitting Fields.
If F, B and E are three fields such that F C B C E, then we
shall refer to B as an intermediate field.
If E is an extension of a field F in which a polynomial p(x) in F
cari be factored into linear factors, and if p(x) cari not be factored
SO
in any intermediate field, then we cal1 E a splitting field for p(x). Thus,
if E is a splitting field of p(x), the roots of p(x) generate E.
A splitting field is of finite degree since it is constructed by a
finite number of adjunctions of algebraic elements, each defining an
extension field of finite degree. Because of the corollary on page 22,
the total degree is finite.
THEOREM 9. If p(x) is a polynomial in a field F, there exists
˜˜
a splitting field E of p(x).
˜˜
We factor p (x ) in F into irreducible factors
f,(x) . . . f*(x) = p(x). If each of these is of the first
degree then F itself is the required splitting field. Suppose
then that fi(x) is of degree higher than the first. By
31
Theorem 7 there is an extension Fr of F in which f r( x ) has a root.
Factor each of the factors f r( x), . . . , fr( x ) into irreducible factors in
Fr and proceed as before. We finally arrive at a field in which p (x)
cari be split into linear factors. The field generated out of F by the
roots of p(x) is the required splitting field.
The following theorem asserts that up to isomorphisms, the
splitting field of a polynomial is unique.
THEOREM 10. Let (T be an isomorphism mapping the field F on
˜˜, Let p (x ) be a polynomial in F and p ™ (x ) the polynomial
the field F™
˜ coefficients corresponding to those of p (x ) under 0. Finally,
in F ™ with
let E be a splitting field of p(x) and E™ a splitting field of p™ (x).
˜
Under these conditions the isomorphism o cari be extended to an
˜˜
isomorphism between E and E™ .
If f(x) is an irreducible factor of p(x) in F, then E contains a
root of f( x ). For let p (x )=(xa J (xa, ) . . (xa .) be the splitting of
p(x) in E. Then (xar)(xa,). . .(xas) = f(x) g(x). We consider
f(x) as a polynomial in E and construct the extension field B = E(a)
= 0. Then(aaI).(aa2)...:(aas) = f(a).g(a) =0
inwhichf(a)
and aai being elements of the field B cari have a product equal to 0
only if f™or one of the factors, say the first, we have aa1 = 0. Thus,
a = al, and a1 is aroot of f(x).
Now in case a11 roots of p(x) are in F, then E = F and p(x)
cari be split in F. This factored form has an image in F™ which is a
splitting, of p™ (x), since the isomorphism o preserves a11 operations
of addition and multiplication in the process of multiplying out the
factors of p(x) and collecting to get the original form. Since p ™ (x)
cari be split in F™ , we must have F ™ = E ™ . In this case, o itself is
the required extension and the theorem is proved if a11 roots of p(x)
are in F.
We proceed by complete induction. Let us suppose the theorem
proved for a11 cases in which the number of roots of p(x) outside of F
is less than n > 1, and suppose that p (x ) is a polynomial having n
roots outside of F. We factor p (x ) into irreducible factors in F;
p(x) = f,(x) fJx) . . f,(x). Not a11 of these factors cari be of
degree 1, since otherwise p (x ) would split in F, contrary to assump
tion. Hence, we may suppose the degree of f 1( x) to be r > 1. Let
f™,(x).f\(x) . . . f;(x) = p™(x) be the factorization of p™(x) into
the polynomials corrrespondng to f 1( x ) , . . . , fm( x ) under O. fi (x )
is irreducible in F ™ , for a factorization of fi (x) in F ™ would induce 1)
under 0l a factorization of f,(x), which was however taken to
be irreducible.
By Theorem 8, the isomorphism o cari be extended to an isomor
phism ol, between the fields F(a) and F ™ (a™ ).
Since F C F(a), p(x) is a polynomial in F(U) and E is a
splitting field for p(x) in F(a). Similarly for p ™ (x). There are now
less than n roots of p (x ) outside the new ground field F (a). Hence
by our inductive assumption o1 cari be extended from an isomorphism
between F(a) and F ™ (a ™ ) to an isomorphism o2 between E and E ™ .
Since u, is an extension of (T, and o2 an extension of o,, we conclude
u2 is an extension of u and the theorem follows.
1) See page 38 for the definition of (2l.
33
Corollary. If p(x) is a polynomial in a field F, then any two
˜
splitting fields for p (x ) are isomorphic.

This follows from Theorem 10 if we take F = F ™ and o to be the
identity mapping, i.e., o(x) = x.
As a consequence of this corollary we see that we are justified
in using the expression ˜?he splitting field of p(x)” since any two
differ only by an isomorphism. Thus, if p (x ) has repeated roots in one
splitting field, SO also in any other splitting field it Will have repeated
roots. The statement “p(x) has repeated roots” Will be significant
without reference to a particular splitting field.
E. Unique Decomposition of Polynomials into Irreducible Factors.

THEOREM 11. If p(x) is a polynomial in a field F, and if

p(x) = pi(x).p,(x)... . .p,(x) = qi(x).q*(x)... . . qs(x) are two
˜
factorizations of p(x) into irreducible polynomials each of degree at
least one, then r = s and after a suitable change in the order in which