# The Fundamental Theorem of Galois Theory

21 Apr

The problem of expressing the roots of a polynomial (in one variable) in terms of its coefficients is an old and fundamental mathematical problem. Work on this problem includes the quadratic formula, the cubic formula, the quartic formula. In ~1830, Evariste Galois created a conceptual framework for understanding the problem in full generality, and as a consequence deduced that the solutions to the general quintic polynomial equation can’t be expressed via nested radicals. In this post we describe Galois’ conceptual framework.

In a previous post, we explained how solving for the roots of a polynomial in terms of its coefficients entails breaking the symmetry present in the coefficients of the polynomial, and we discussed how to break this symmetry in the special case of a quadratic polynomial. In a subsequent post, we characterized the polynomial expressions of the coefficients as precisely the polynomial expressions in the roots that are invariant under all permutations of the roots.

In the case of a quadratic polynomial, the only symmetry that needs to be broken is invariance under swapping of the two roots. In the case of a generic polynomial of degree ‘n,’ it’s necessary to pass from expressions invariant under all permutations of the n roots to an expression that’s not invariant under any nontrivial permutation of the n roots.

The strategy for doing this is to break the symmetry a little bit at a time: starting from expressions that are invariant under all permutations and successively passing to expressions that are invariant under fewer and fewer permutations, finally ending up with an expression that isn’t invariant under any nontrivial permutations of the roots.

To codify this perspective, we introduce the Galois group of a polynomial.

Let P(x) be a polynomial with rational coefficients. Let K be the smallest number system containing the roots of P(x) that is closed under addition, subtraction, multiplication and division. Then K is called the splitting field of P(x). The splitting field of P(x) can also be defined as the collection of multivariable polynomial expressions in the roots of P(x), with rational coefficients.

The idea then is to classify elements of K according to how symmetric they are with respect to permutations of the roots, with a view toward developing a hierarchy of subsets of K from “least symmetric” to “most symmetric” and then think about the different levels on the hierarchy, and their relations. To this end, we consider a collection of permutations of the roots of P(x) defined by Evariste Galois. Giving the definition will take a few paragraphs, and its utility is not immediately apparent, but it will be vindicated by the end of this post.

An automorphism of K is a bijection from K to K such that

$\large \phi(a + b) = \phi(a) + \phi(b), \hspace{0.3 cm} \phi(ab) = \phi(a)\phi(b)$

The reader can check that these properties imply that an automorphism of K leaves the rational numbers invariant. The action of an automorphism on the other elements of K can be described as follows. An element of K can be written as

$\large R(r_1, r_2, ... r_n)$

where R is a multivariable polynomial with rational coefficients. The reader can check that the definition of automorphism implies that

$\large \phi(Q(r_1, r_2, \ldots r_n)) = Q(\phi(r_1), \phi(r_2), \ldots, \phi(r_n))$

Thus, an automorphism of K is determined by where it sends the roots of P(x). Where a root of P(x) goes is heavily restricted: if

$\large P(r) = 0$

then the properties of an automorphism imply that

$\large \phi(P(r)) = P(\phi(r)) = \phi(0) = 0$

so that an automorphism of K sends roots of P(x) to roots of P(x). Since automorphisms are bijective (by definition), they must induce permutations of the set of roots of P(x). Thus, an automorphism of K can represented by a permutation of the roots of P(x).

None of this says anything about which permutations of the roots of P(x) induce automorphisms of K, something which is dependent on the particular polynomial. In a later post, we’ll discuss this subject in detail. For most choices of P(x), every permutation induces an automorphism. But there are exceptional polynomials P(x) for which there are permutations that don’t induce automorphisms.

A collection of permutations is called a group if it’s closed under composition of permutations. Successive application of automorphisms yields another automorphism, so the set of [permutations corresponding to automorphisms of K] form a group, called the Galois Group of P(x) (or of K). This is denoted Gal(K/Q) (where Q is the rational numbers), but for simplicity, we’ll write it as G.

We now return to the idea of classifying elements of K by the permutations that leave them invariant:

Given an element ‘y’ of K, the set of elements of G that leave ‘y’ invariant forms a subgroup H of G. We want to classify elements of K by these subgroups of G. The collection of elements of K that are invariant under H is closed under addition, subtraction, multiplication and division, and so is called a subfield of K, which is referred to as a fixed field of H.

The Fundamental Theorem of Galois Theory: Every subfield of K is a fixed field of a unique subgroup of G.

A special case of the theorem, which is the key to its proof, is the case where the subgroup H consists of all of G:

Lemma: The elements of K that are invariant with respect to G are precisely the rational numbers.

The fact that the rational numbers are invariant with respect to G was mentioned earlier in the post: the substantive content of the lemma is that if an element ‘y’ of K is invariant with respect to G, then it’s a rational number.

Because P(x) has rational coefficients, Newton’s theorem on symmetric polynomials implies that if ‘y’ is invariant with respect to all permutations of the roots, then it’s a rational number. The lemma characterizes the smallest subgroup of H permutations of the roots such that [if ‘y’ is invariant under H, ‘y’ is a rational number] as being the Galois group of K.

Thus, we see that the Galois group holds the key to classifying all elements of K according to the permutations under which they’re invariant. This allows one to study the problem of breaking symmetry to find expressions for the roots of a polynomial in terms of its coefficients in a systematic way.

In future posts, we’ll pursue this line of thought further, as well as proving the lemma, and giving concrete examples of splitting fields, Galois groups, subgroups of Galois groups, the fixed fields of these subgroups, etc.