Normal extension


In abstract algebra, a normal extension is an algebraic field extension L/K for which every polynomial that is irreducible over K either has no root in L or splits into linear factors in L. Bourbaki calls such an extension a quasi-Galois extension.

Definition

The algebraic field extension L/K is normal if every irreducible polynomial over K that has at least one root in L splits over L. In other words, if α ∈ L, then all conjugates of α over K belong to L.

Equivalent properties

The normality of L/K is equivalent to either of the following properties. Let Ka be an algebraic closure of K containing L.
If L is a finite extension of K that is separable then the following property is also equivalent:
Let L be an extension of a field K. Then:
For example, is a normal extension of since it is a splitting field of On the other hand, is not a normal extension of since the irreducible polynomial has one root in it, but not all of them. Recall that the field of algebraic numbers is the algebraic closure of i.e., it contains Since,
and, if ω is a primitive cubic root of unity, then the map
is an embedding of in whose restriction to is the identity. However, σ is not an automorphism of.
For any prime, the extension is normal of degree. It is a splitting field of. Here denotes any th primitive root of unity. The field is the normal closure of.

Normal closure

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is a normal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i.e., the only subfield of M which contains L and which is a normal extension of K is M itself. This extension is called the normal closure of the extension L of K.
If L is a finite extension of K, then its normal closure is also a finite extension.