In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.

Suppose that ${\displaystyle E}$ is an extension of the field ${\displaystyle F}$ (written as ${\displaystyle E/F}$ and read "E over F"). An automorphism of ${\displaystyle E/F}$ is defined to be an automorphism of ${\displaystyle E}$ that fixes ${\displaystyle F}$ pointwise. In other words, an automorphism of ${\displaystyle E/F}$ is an isomorphism ${\displaystyle \alpha :E\to E}$ such that ${\displaystyle \alpha (x)=x}$ for each ${\displaystyle x\in F}$. The set of all automorphisms of ${\displaystyle E/F}$ forms a group with the operation of function composition. This group is sometimes denoted by ${\displaystyle \operatorname {Aut} (E/F).}$

If ${\displaystyle E/F}$ is a Galois extension, then ${\displaystyle \operatorname {Aut} (E/F)}$ is called the Galois group of ${\displaystyle E/F}$, and is usually denoted by ${\displaystyle \operatorname {Gal} (E/F)}$.

If ${\displaystyle E/F}$ is not a Galois extension, then the Galois group of ${\displaystyle E/F}$ is sometimes defined as ${\displaystyle \operatorname {Aut} (K/F)}$, where ${\displaystyle K}$ is the Galois closure of ${\displaystyle E}$.

Another definition of the Galois group comes from the Galois group of an irreducible polynomial ${\displaystyle f\in F[x]}$. If there is a field ${\displaystyle K/F}$ such that ${\displaystyle f}$ factors as a product of distinct linear polynomials

over the field ${\displaystyle K}$, then the Galois group of the polynomial ${\displaystyle f}$ is defined as the Galois group of ${\displaystyle K/F}$ where ${\displaystyle K}$ is minimal among all such fields.

One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension ${\displaystyle K/k}$, there is a bijection between the set of subfields ${\displaystyle k\subset E\subset K}$ and the subgroups ${\displaystyle H\subset G.}$ Then, ${\displaystyle E}$ is given by the set of invariants of ${\displaystyle K}$ under the action of ${\displaystyle H}$, so