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 integers or the rational numbers), there exists

• a field L that contains K (or a field isomorphic to K) as a subfield, and also the roots of $\alpha$.
• a group containing all automorphisms in L that leave the elements in K untouched.

Providing certain technicalities are fullfilled, the structure of this group contains information about the nature of the roots, and whether the equation $\alpha$ has solutions expressible as radical expressions - i.e. formulas involving a simple sequence of ordinary arithmetical expressions and rational powers.