Normal extension - Definition and Overview

In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X].

The following conditions are equivalent to L/K being a normal extension:

• Let K^a an algebraic closure of K containing L. Every embedding σ of L in K^a such that σ restricts to the identity on K, verifies σ(L)=L. In other words, σ is an automophism of L over K.

• Every irreducible polynomial in K[X] which has a root in L factors into linear factors in L[X].

For example, Q(√2)/Q is a normal extension, but Q(⁴√2)/Q is not a normal extension since it is missing some roots of X⁴-2.