Wikipedia / Normal extension
Wikipedia / 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 (i.e., L is an algebraic extension of K (i.e., L is contained in an algebraic closure of K). Then the following conditions, any of which can be regarded as a definition of normal extension, are equivalent:
Every embedding of L over K induces an automorphism of L.
L is the splitting field.
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 (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
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;
of polynomials that each splits over L are fields, then S has a polynomial that does not split in F;
that fix all elements of K have the same image;
of L that fix all elements of K that fix all elements of K.
Examples and counterexamples
), but not all of them (it does not have the non-real cubic roots of of algebraic numbers
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.