In abstract algebra, a normal extension is an algebraic field extension L/K for which every irreducible polynomial over K which has a root in L, splits into linear factors in L. These are one of the conditions for algebraic extensions to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension.
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, which any of them can be regarded as a definition of normal extension, are equivalent:
- Every embedding of L in induces an automorphism of L.
- L is the splitting field of a family of polynomials in .
- Every irreducible polynomial of which has a root in L splits into linear factors in L.
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 (that is, 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.
Equivalent conditions for normality
Let be algebraic. The field L is a normal extension if and only if any of the equivalent conditions below hold.
- The minimal polynomial over K of every element in L splits in L;
- There is a set of polynomials that simultaneously split over L, such that if are fields, then S has a polynomial that does not split in F;
- All homomorphisms have the same image;
- The group of automorphisms, of L which fixes elements of K, acts transitively on the set of homomorphisms
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 that is, it contains Since,
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
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, that is, 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.