Galois Groups

In Algebraic Numbers we discussed algebraic number fields and a very important group associated associated to an algebraic number field called its ideal class group. In this post we define another very important group called the Galois group. They are named after the mathematician Evariste Galois, who lived in the early 19th century and developed the theory before his early death in a duel (with mysterious circumstances) at the age of 20 years old.

The problem that motivated the development of Galois groups was the solution of polynomial equations of higher degree. We know that for quadratic equations (equations of degree 2) there exists a “quadratic formula” that allows us to solve for the roots of any quadratic equation. For cubic equations (equations of degree 3) and quartic equations (equations of degree 4), there is also a similar “cubic formula” and a “quartic formula”, although they are not as well-known as the  quadratic formula.

However for quintic equations (equations of degree 5) there is no “quintic formula”. What this means is that not every quintic equation can be solved by a finite number of additions, subtractions, multiplications, divisions, and extractions of roots. Some quintic equations, of course, can be easily solved using these operations, such as x^{5}-1=0. However this does not hold true for all quintic equations. This was proven by another mathematician, Niels Henrik Abel, but it was Galois who gave the conditions needed to determine whether a quintic equation could be solved using the aforementioned operations or not.

The groundbreaking strategy that Galois employed was to study the permutations of roots of polynomial equations. These permutations are the same as the field automorphisms of the smallest field extension (see Algebraic Numbers for the definition of a field extension) that contains these roots (called the splitting field of the polynomial equation) which also fix the field of coefficients of the polynomial.

By “field automorphisms” we mean a function f from a field to itself such that the following conditions are satisfied:

f(a+b)=f(a)+f(b)

f(ab)=f(a)f(b)

By “fix” we mean that if a is an element of the field of coefficients of the polynomial equation, then we must have

f(a)=a

We might perhaps do better by discussing an example. We do not delve straight into quintic equations, and consider first the much simpler case of a quadratic equation such as x^{2}+1=0. We consider the polynomial x^{2}+1 as having coefficients in the field \mathbb{Q} of rational numbers. The roots of this equation are i and -i, and the splitting field is the field \mathbb{Q}[i].

Since there are only two roots, we only have two permutations of these roots. One is the identity permutation, which sends i to i and -i to -i, and the other is the permutation that exchanges the two, sending i to -i and -i to i. The first one corresponds to the identity field automorphism of \mathbb{Q}[i], while the second one corresponds to the complex conjugation field automorphism of \mathbb{Q}[i]. Both these permutations preserve \mathbb{Q}.

These permutations (or field automorphisms) form a group (see Groups), which is what we refer to as the Galois group of the field extension (the splitting field, considered as a field extension of the field of coefficients of the polynomial) or the polynomial.

The idea is that the “structure” of the Galois group, as a group, is related to the “structure” of the field extension. For example, the subgroups of the Galois groups correspond to the “intermediate fields” contained in the splitting field but containing the field of coefficients of the polynomial.

Using this idea, Galois showed that whenever the Galois group of an irreducible quintic polynomial is the symmetric group S_{5} (the group of permutations of the set with 5 elements) or the alternating group A_{5} (the group of “even” permutations of the set with 5 elements), then the polynomial cannot be solved using a finite number of additions, subtractions, multiplications, division, and extractions of roots. This happens, for example, when the irreducible quintic polynomial has three real roots, as in the case of x^{5}-16x+2. More details of the proof can be found in the last chapter of the book Algebra by Michael Artin.

Although the Galois group was initially developed to deal with problems regarding the solvability of polynomial equations, they have found applications beyond this original purpose and have become a very important part of many aspects of modern mathematics, especially in (but not limited to, rather surprisingly) number theory.

For example, the study of “representations” of Galois groups in terms of linear transformations of vector spaces (see Vector Spaces, Modules, and Linear Algebra) is an important part of the proof of the very famous problem called Fermat’s Last Theorem by the mathematician Andrew Wiles in 1994. A very active field of research in the present day related to representations of Galois groups is called the Langlands program. In particular, what is usually being studied is the “absolute” Galois group – the group of field automorphisms of the set of all algebraic numbers that fix the field \mathbb{Q} of rational numbers. A book that makes these ideas accessible to a more general audience is Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross.

References:

Galois Theory on Wikipedia

Galois Group on Wikipedia

Wiles’ Proof of Fermat’s Last Theorem on Wikipedia

Langlands Program on Wikipedia

Fearless Symmetry: Exposing the Hidden Patterns of Numbers by Avner Ash and Robert Gross

Algebra by Michael Artin

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