Let be an algebraic extension (i.e., L is an algebraic extension of K), such that (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:[3]
There is a set of polynomials that each splits over L, such that if are fields, then S has a polynomial that does not split in F;
All homomorphisms that fix all elements of K have the same image;
The group of automorphisms, of L that fix all elements of K, acts transitively on the set of homomorphisms that fix all elements of K.
Examples and counterexamples
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 (namely, ), but not all of them (it does not have the non-real cubic roots of 2). Recall that the field of algebraic numbers is the algebraic closure of and thus it contains Let be a primitive cubic root of unity. Then since,
the map
is an embedding of in whose restriction to is the identity. However, 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 (see below) 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 that is minimal, that is, the only subfield of M that contains L and that 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.