Normal Extension - Equivalent Properties and Examples

Equivalent Properties and Examples

The normality of L/K is equivalent to each of the following properties:

• Let Ka be an algebraic closure of K containing L. Every embedding σ of L in Ka which restricts to the identity on K, satisfies σ(L) = L. In other words, σ is an automorphism of L over K.
• Every irreducible polynomial in K which has a root in L factors into linear factors in L.
• The minimal polynomial over K of every element in L splits over L.

For example, is a normal extension of, since it is a splitting field of x2 − 2. On the other hand, is not a normal extension of since the polynomial x3 − 2 has one root in it (namely, ), but not all of them (it does not have the non-real cubic roots of 2).

The fact that is not a normal extension of can also be proved using the first of the two equivalent properties from above. The field of complex algebraic numbers is an algebraic closure of containing . On the other hand

and, if ω is one of the two non-real cubic roots of 2, then the map

is an embedding of in whose restriction to is the identity. However, σ is not an automorphism of .

For any prime p, the extension is normal of degree p(p − 1). It is a splitting field of xp − 2. Here denotes any pth primitive root of unity.