Normal extension


In abstract algebra, a normal extension is an algebraic field extension L/''K for which every irreducible polynomial over K'' that has a root in L splits into linear factors over L. This is one of the conditions for an algebraic extension to be a Galois extension. Bourbaki calls such an extension a quasi-Galois extension. For finite extensions, a normal extension is identical to a splitting field.

Definition

Let ' be an algebraic extension, such that . Then the following conditions, any of which can be regarded as a definition of normal extension', are equivalent:
  • Every embedding of L'' in over K induces an automorphism of L.
  • L is the splitting field of a family of polynomials in.
  • Every irreducible polynomial of that has a root in L splits into linear factors in L.

    Other properties

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, 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 each splits over L, such that if are fields, then S has a polynomial that does not split in F;
  • All homomorphisms that fix all elements of K have the same image;
  • The group of automorphisms, of L that fix all elements of K, acts transitively on the set of homomorphisms that fix all elements 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, but not all of them. Recall that the field of algebraic numbers is the algebraic closure of and thus it contains Let be a primitive cubic root of unity. Then since,
the map
is an embedding of in whose restriction to is the identity. However, is not an automorphism of
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 of

Normal closure

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 that is minimal, that is, the only subfield of M that contains L and that 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.