Galois theory

Galois theory relates field extensions to groups of automorphisms. For a finite Galois extension $L/K$, its Galois group consists of the field automorphisms of $L$ fixing $K$. The Fundamental theorem of Galois theory matches intermediate fields with subgroups, reversing inclusion.

For example, $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$ has four automorphisms, independently changing the two square-root signs. The subgroup fixing $\sqrt{2}$ corresponds to $\mathbb{Q}(\sqrt{2})$. This translates field questions into finite symmetry questions. Applied to a splitting field, it characterizes when a polynomial equation is solvable by radicals: the corresponding Galois group must be a solvable group.

Galois theory is a branch of abstract algebra that studies the relationships between field extensions and group theory, particularly focusing on the solvability of polynomial equations. Named after the mathematician Évariste Galois, it provides a powerful framework for understanding how the roots of polynomials are related to the symmetry properties of the equations. The core ideas of Galois theory can be summarized as follows: 1. **Field Extensions**: A field extension is a bigger field that contains a smaller field.