Normal extension

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, any of which 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. 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 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, and, if is a primitive cubic root of unity, then the map is an embedding of in whose restriction to is the identity.