Section 5.6 The Elements of Galois Theory

Given a polynomial in , the polynomial ring in over , we shall associate with a group, called the Galois group of . There is a very close relationship between the roots of a polynomial and its Galois group; in fact, the Galois group will turn out to be a certain permutation group of the roots of the polynomial. We shall make a study of these ideas in this, and in the next, section.

The means of introducing this group will be through the splitting field of over , the Galois group of being defined as a certain group of automorphisms of this splitting field. This accounts for our concern, in so many of the theorems to come, with the automorphisms of a field. A beautiful duality, expressed in the fundamental theorem of the Galois theory (Theorem 5.6.6), exists between the subgroups of the Galois group and the subfields of the splitting field. From this we shall eventually derive a condition for the solvability by means of radicals of the roots of a polynomial in terms of the algebraic structure of its Galois group. From this will follow the classical result of Abel that the general polynomial of degree 5 is not solvable by radicals. Along the way we shall also derive, as side results, theorems of great interest in their own right. One such will be the fundamental theorem on symmetric functions. Our approach to the subject is founded on the treatment given it by Artin.

Recall that we are assuming that all our fields are of characteristic , hence we can (and shall) make free use of Theorem 5.5.1 and its corollary.

By an automorphism of the field we shall mean, as usual, a mapping of onto itself such that

for all . Two automorphisms and of are said to be distinct if for some element in .

We begin the material with

THEOREM 5.6.1: If is a field and if are distinct automorphisms of , then it is impossible to find elements , not all in such that

for all .

group of automorphisms of , then the fixed field of is the set of all elements such that for all .

Note that this definition makes perfectly good sense even if is not a
group but is merely a set of automorphisms of . However, the fixed field of a set of automorphisms and that of the group of automorphisms generated by this set (in the group of automorphisms of ) are equal (Problem 1), hence we lose nothing by defining the concept just for groups of automorphisms. Besides, we shall only be interested in the fixed fields of group of automorphisms.

Having called the set, in the definition above, the fixed
field of , it would be nice if this terminology were accurate. That it is we see in

LEMMA 5.6.1: The fixed
field of is a subfield of .

We shall be concerned with the automorphisms of a
field which behave in a prescribed manner on a given subfield.

field and let be a subfield of . Then the group of automorphisms of relative to , written , is the set of all automorphisms of leaving every element of fixed; that , the automorphisms of is in if and only if for every .

It is not surprising, and is quite easy to prove

LEMMA 5.6.2: is a
subgroup of the group of all automorphisms of .

We leave the proof of this lemma to the reader. One remark: contains the field of rational numbers , since is of characteristic , and it is easy to see that the fixed field of any group of automorphisms of , being a field, must contain . Hence, every rational number is left fixed by every automorphisms of .

We pause to examine a few examples of the concepts just introduced.

The examples, although illustrative, are still too special, for note that in each of them turned out to be a
cyclic group. This is highly atypical for, in general, need not even be abelian (see Theorem 5.6.3). However, despite their specialty, they do bring certain important things to light. For one thing they show that we must study the effect of the automorphisms on the roots of polynomials and, for another, they point out that need not be equal to all of the fixed field of . The cases in which this does happen are highly desirable ones and are situations with which we shall soon spend much time and effort.

THEOREM 5.6.2: If is a finite extension of , then is a finite group and its order,   satisfies .

Theorem 5.6.2 is of central importance in the
Galois theory. However, aside from its key role there, it serves us well in proving a classic result concerned with symmetric rational functions. This result on symmetric functions in its turn will play an important part in the Galois theory.

First a few remarks on the field of rational functions in -variables over a field . Let us recall that in Section 3.11 we defined the ring of polynomials in the -variables over and from this defined the field of rational functions in , over as the ring of all quotients of such polynomials.

Let be the symmetric group of degree considered to be acting on the set ; for and an integer with , let be the image of under . We can make act on in the following natural way: for and , define the mapping which takes onto . We shall write this mapping of onto itself also as . It is obvious that these mappings define automorphisms of . What is the fixed field of with respect to ? It consists of all rational functions such that

for all . But these are precisely those elements in which are known as the
symmetric rational functions. Being the fixed field of they form a subfield of , called the field of symmetric rational functions which we shall denote by . We shall be concerned with three questions:

1. What is ?

2. What is )?

3. Can we describe in terms of some particularly simple
extension of ?

We shall answer these three questions simultaneously.

We can explicitly produce in some particularly simple functions constructed from known as the
elementary symmetric functions in . These are defined as follows:

That these are
symmetric functions is left as an exercise. For and 4 we write them out explicitly below.

Note that when and are the roots of the polynomial

that when , and are roots of

and that when , and are all roots of

Since are all in , the field obtained by adjoining to must lie in . Our objective is now twofold, namely, to prove

1. .

2. .

Since the group is a group of automorphisms of leaving fixed,

Thus, by Theorem 5.6.2,

If we could show that

then, since is a subfield of , we would have

But then we would get that

and so , and, finally,

(this latter from the second sentence of this paragraph). These are precisely the conclusions we seek.

Thus, we merely must prove that

To see how this settles the whole affair, note that the polynomial

which has coefficients in , factors over as

(This is in fact the origin of the
elementary symmetric functions.) Thus , of degree over , splits as a product of linear factors over . It cannot split over a proper subfield of which contains for this subfield would then have to contain both and each of the roots of , namely, ; but then this subfield would be all of . Thus we see that is the splitting field of the polynomial

over . Since is of degree , by Theorem 5.3.2 we get

Thus all our claims are established. We summarize the whole discussion in the basic and important result

THEOREM 5.6.3: Let be a field and let be the field of rational functions in   over . Suppose that is the field of
symmetric rational functions; then

1. .

2. , the
symmetric group of degree .

3. If are the
elementary symmetric functions in , then

4. is the splitting field over of the polynomial

We mentioned earlier that given any integer it is possible to construct a field and a polynomial of degree over this field whose splitting field is of maximal possible degree, , over this field. Theorem 5.6.3 explicitly provides us with such an example for if we put , the rational function field in variables and consider the splitting field of the polynomial over then it is of degree ! over .

Part 3 of Theorem 5.6.3 is a very classical theorem. It asserts that a
symmetric rational function in variables is a rational function in the elementary symmetric functions of these variables. This result can even be sharpened to: A symmetric polynomial in variables is a polynomial in their elementary symmetric functions (see Problem 7). This result is known as the theorem on symmetric polynomials.

In the examples we discussed of groups of automorphisms of fields and of fixed fields under such groups, we saw that it might very well happen that is actually smaller than the whole fixed field of . Certainly is always contained in this field but need not fill it out. Thus to impose the condition on an extension of that be precisely the fixed field of is a genuine limitation on the type of extension of that we are considering. It is in this kind of extension that we shall be most interested.

DEFINITION: is a normal extension of if is a finite extension of such that is the fixed field of .

Another way of saying the same thing: If is a normal extension of , then every element in which is outside is moved by some element in . In the examples discussed, Examples 5.6.1 and 5.6.3 were normal extensions whereas Example 5.6.2 was not.

An immediate consequence of the assumption of normality is that it allows us to calculate with great accuracy the size of the fixed field of any subgroup of and, in particular, to sharpen Theorem 5.6.2 from an inequality to an equality.

THEOREM 5.6.4: Let be a normal extension of and let be a subgroup of ; let

be the fixed field of H. Then

1.  .

2.  .

(In particular, when .)

When , by the normality of over ; consequently for this particular case we read off the result .

We are rapidly nearing the central theorem of the Galois theory. What we still lack is the relationship between splitting fields and normal extensions. This gap is filled by

THEOREM 5.6.5: is a
normal extension of if and only if is the splitting field of some polynomial over .

DEFINITION: Let be a polynomial in and let be its splitting field over . The Galois group of is the group of all the automorphisms of , leaving every element of fixed.

Note that the of can be considered as a group of permutations of its roots, for if is a root of and if then is also a root of .

We now come to the result known as the fundamental theorem of Galois theory. It sets up a one-to-one correspondence between the subfields of the splitting field of and the subgroups of its Galois group. Moreover, it gives a criterion that a subfield of a normal extension itself be a normal extension of . This fundamental theorem will be used in the next section to derive conditions for the solvability by radicals of the roots of a polynomial.

THEOREM 5.6.6: Let be a polynomial in its splitting field over , and its Galois group. For any subfield of which contains let

and for any subgroup of let

Then the association of with sets up one-to-one correspondence of the set of subfields of which contain onto the set of subgroups of such that

1. .

2. .

3. = index of   in .

4. is a normal extension of if and only if is a normal subgroup of .

5. When is a normal extension of , then is isomorphic to