Galois field extension

This term is related to: Galois theory
View other terms related to Galois theory | View facts related to Galois theory

Definition

• A field extension $L/K$ is called Galois if it is algebraic and $K=L^{\mathrm{A}\mathrm{u}\mathrm{t}(L/K)}$. That is, if $x\in L$ and $x$ is fixed by all automorphisms of $L$ fixing $K$, then in fact $x\in K$.

When the field extension is finite

Further information: Equivalence of definitions of finite Galois field extension

When $L/K$ is finite, the following are equivalent to the given statement, and may be used as a definition of a Galois extension:

• A finite field extension $L/K$ is called Galois if it is normal and separable.

• A finite field extension $L/K$ is called Galois if $L$ is the splitting field of a separable polynomial with coefficients in $K$.

Examples

The field extension $\mathbb{C}/\mathbb{R}$ is Galois. Indeed, the automorphisms of $\mathbb{C}$ fixing $\mathbb{R}$ are precisely the identity map and complex conjugation. Indeed, if $z\in \mathbb{C}$, $z\in \mathbb{R}$ if and only if it is fixed by these two automorphisms. Thus $\mathbb{C}/\mathbb{R}$ is Galois (with Galois group cyclic of order 2.)