equivalent conditions for normality of a field extension
(1)(2) Let be an -basis for , and for each , let be the irreducible polynomial of over . By hypothesis, each splits over , and because we evidently have , it follows that is a splitting field of over .
(2)(3) Assume that is a splitting field over of . Given , we may write for some ; because fixes pointwise, we have for , and since is injective, it must simply permute the roots of . Thus . As is generated over by the roots of the polynomials in , we obtain .
(3)(1) Let be an algebraic closure of , noting that, since is algebraic over , that same is true of , and consequently is an algebraic closure of containing . Now suppose is irreducible and that is a root of , and let be any root of in . There exists an -isomorphism such that . Because is a splitting field over both and of the set of irreducible polynomials in , extends to an -isomorphism . It follows that is an -monomorphism, so that, by hypothesis, , hence that . Thus splits over , and therefore is normal. ∎
|Title||equivalent conditions for normality of a field extension|
|Date of creation||2013-03-22 18:37:56|
|Last modified on||2013-03-22 18:37:56|
|Last modified by||azdbacks4234 (14155)|