Galois extension

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.

The property of an extension being Galois behaves well with respect to field composition and intersection.

Characterization of Galois extensions

An important theorem of Emil Artin states that for a finite extension ${\displaystyle E/F}$, each of the following statements is equivalent to the statement that ${\displaystyle E/F}$ is Galois:

• ${\displaystyle E/F}$ is a normal extension and a separable extension.
• ${\displaystyle E}$ is a splitting field of a set of separable polynomials with coefficients in ${\displaystyle F}$.
• ${\displaystyle |\mathrm{Aut}(E/F)|=[E:F]}$, that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

• Every irreducible polynomial in ${\displaystyle F[x]}$ with at least one root in ${\displaystyle E}$ splits over ${\displaystyle E}$ and is separable.
• ${\displaystyle |\mathrm{Aut}(E/F)|\ge [E:F]}$, that is, the number of automorphisms is at least the degree of the extension.
• ${\displaystyle F}$ is the fixed field of a subgroup of ${\displaystyle \mathrm{Aut}(E)}$.
• ${\displaystyle F}$ is the fixed field of ${\displaystyle \mathrm{Aut}(E/F)}$.
• There is a one-to-one correspondence between subfields of ${\displaystyle E/F}$ and subgroups of ${\displaystyle \mathrm{Aut}(E/F)}$.

An infinite field extension ${\displaystyle E/F}$ is Galois if and only if ${\displaystyle E}$ is the union of finite Galois subextensions ${\displaystyle E_i/F}$ indexed by an (infinite) index set ${\displaystyle I}$, i.e. ${\displaystyle E=\bigcup _{i\in I}{E}_i}$ and the Galois group is an inverse limit ${\displaystyle \mathrm{Aut}(E/F)=\underset{i\in I}{\underleftarrow{\lim }}\mathrm{Aut}(E_i/F)}$ where the inverse system is ordered by field inclusion ${\displaystyle E_i\subset E_j}$.