Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable;^[1] or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.^[a]

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.^[2]

The property of an extension being Galois behaves well with respect to field composition and intersection.^[3]

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

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

An infinite field extension ${\displaystyle E/F}${\displaystyle E/F} is Galois if and only if ${\displaystyle E}${\displaystyle E} is the union of finite Galois subextensions ${\displaystyle E_{i}/F}${\displaystyle E_{i}/F} indexed by an (infinite) index set ${\displaystyle I}${\displaystyle I}, i.e. ${\displaystyle E=\bigcup _{i\in I}E_{i}}${\displaystyle E=\bigcup _{i\in I}E_{i}} and the Galois group is an inverse limit ${\displaystyle \operatorname {Aut} (E/F)=\varprojlim _{i\in I}{\operatorname {Aut} (E_{i}/F)}}${\displaystyle \operatorname {Aut} (E/F)=\varprojlim _{i\in I}{\operatorname {Aut} (E_{i}/F)}} where the inverse system is ordered by field inclusion ${\displaystyle E_{i}\subset E_{j}}${\displaystyle E_{i}\subset E_{j}}.^[4]

There are two basic ways to construct examples of Galois extensions.

• Take any field ${\displaystyle E}${\displaystyle E}, any finite subgroup of ${\displaystyle \operatorname {Aut} (E)}${\displaystyle \operatorname {Aut} (E)}, and let ${\displaystyle F}${\displaystyle F} be the fixed field.
• Take any field ${\displaystyle F}${\displaystyle F}, any separable polynomial in ${\displaystyle F[x]}${\displaystyle F[x]}, and let ${\displaystyle E}${\displaystyle E} be its splitting field.

Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cubic root of 2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first of them is the splitting field of ${\displaystyle x^{2}-2}${\displaystyle x^{2}-2}; the second has normal closure that includes the complex cubic roots of unity, and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in the real numbers and ${\displaystyle x^{3}-2}${\displaystyle x^{3}-2} has just one real root. For more detailed examples, see the page on the fundamental theorem of Galois theory.

An algebraic closure ${\displaystyle {\bar {K}}}${\displaystyle {\bar {K}}} of an arbitrary field ${\displaystyle K}${\displaystyle K} is Galois over ${\displaystyle K}${\displaystyle K} if and only if ${\displaystyle K}${\displaystyle K} is a perfect field.

• Galois theory
• Algebraic number theory
• Field extensions

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