Galois theory

Galois theory is an area of mathematical study that originated with Evariste Galois around 1830, as part of an effort to understand the relationships between the roots of polynomials, in particular why there are no simple formulas for extracting the roots of the general polynomial of fifth (or higher) degree.

Introduction

Galois expressed his theory in terms of polynomials and complex numbers, today Galois theory is usually formulated using general field theory.

Key concepts are field extensions and groups, which should be thoroughly understood before Galois theory can be properly studied.

The core idea behind Galois theory is that given a polynomial $\alpha$ with coefficients in a field K (typically the rational numbers), there exists

• a smallest possible field L that contains K (or a field isomorphic to K) as a subfield and also all the roots of $\alpha$. This field is known as the extension of K by the roots of $\alpha$.
• a group containing all automorphisms in L that leave the elements in K untouched - the Galois group of the polynomial $\alpha$.

Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation $\alpha =0$ has solutions expressible as a finite formula involving only ordinary arithmetical operations (addition, subtraction, multiplication, division and rational powers) on the coefficients.

The Galois group of a polynomial - a basic example

As an example, let us look at the second-degree polynomial $x^{2}-5$, with the coefficients {-5,0,1} viewed as elements of Q.

This polynomial has no roots in Q. However, from the fundamental theorem of algebra we know that it has exacly two roots in C, and can be written as the product of two first-degree polynomials there - i.e. $x^{2}-5=(x-r_{0})(x-r_{1}),r_{0},r_{1}\in C$. From direct inspection of the polynomial we also realize that $r_{0}=-r_{1}$.

We now look for the smallest subfield of C that contains Q and both $r_{0}$ and $-r_{0}$, which is L = { $a+br_{0},a,b\in Q$ }. Since $r_{0}^{2}=5\in Q$, all products and sums are well defined. This field is then the smallest extension of Q by the roots of $\alpha$.

Now, in order to find the Galois group, we need to look at all possible automorphisms of L that leave every elements of Q alone.

The only such automorphisms are the null automorphism and the map $a+br_{0}\rightarrow a-br_{0}$.

Under composition of automorphisms, these two automorphisms together are isomorphic to the group $S_{2}$, the group of permutations of two objects.

The sought for Galois group is therefore $S_{2}$.