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.
The algebraic field extension L/K is normal (we also say that L is normal over K) 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 (i.e., all roots of the minimal polynomial of α over K) belong to L.
The normality of L/K is equivalent to either of the following properties. Let Ka be an algebraic closure of K containing L.
- Every embedding σ of L in Ka that restricts to the identity on K, satisfies σ(L) = L (σ is an automorphism of L over K.)
- 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.)
If L is a finite extension of K that is separable (for example, this is automatically satisfied if K is finite or has characteristic zero) then the following property is also equivalent:
- There exists an irreducible polynomial whose roots, together with the elements of K, generate L. (One says that L is the splitting field for the polynomial.)
Let L be an extension of a field K. Then:
- If L is a normal extension of K and if E is an intermediate extension (i.e., L ⊃ E ⊃ K), then L is a normal extension of E.
- If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normal extensions 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 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 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. The field is the normal closure (see below) of .
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.