Normal Extension

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].

Equivalent Properties
The normality of L/K is equivalent to either of the following properties. Let K^a be an algebraic closure of K containing L.

1) Every embedding \sigma of L in K^a that restricts to the identity on K, satisfies \sigma(L)=L. In other words, \sigma, is an automorphism of L over K.
2) Every irreducible polynomial in K[X] that has one root in L, has all of its roots in L, that is, it decomposes into linear factors in L[X]. (One says that the polynomial splits in L.)

Author: mathtuition88

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