Field extension
Normal, separable and Galois extensions

An algebraic extension L / K {\displaystyle L/K} is called normal if every irreducible polynomial in K that has a root in L completely factors into linear factors over L. Every algebraic extension F / K admits a normal closure L, which is an extension field of F such that L / K {\displaystyle L/K} is normal and which is minimal with this property.

An algebraic extension L / K {\displaystyle L/K} is called separable if the minimal polynomial of every element of L over K is separable, i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a field extension that is both normal and separable.

A consequence of the primitive element theorem states that every finite separable extension has a primitive element (i.e. is simple).

Given any field extension L / K {\displaystyle L/K}, we can consider its automorphism group Aut (L / K) {\displaystyle {\text{Aut}}(L/K)}, consisting of all field automorphisms α: L → L with α (x) = x for all x in K. When the extension is Galois this automorphism group is called the Galois group of the extension. Extensions whose Galois group is abelian are called abelian extensions.

For a given field extension L / K {\displaystyle L/K}, one is often interested in the intermediate fields F (subfields of L that contain K). The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediate fields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by the fundamental theorem of Galois theory.
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