equivalent conditions for normality of a field extension


If K/F is an algebraic extensionMathworldPlanetmath of fields, then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.

    K is normal over F;

  2. 2.

    K is the splitting fieldMathworldPlanetmath over F of a set of polynomialsMathworldPlanetmathPlanetmathPlanetmath in F[X];

  3. 3.

    if F¯ is an algebraic closureMathworldPlanetmath F containing K and σ:KF¯ is an F-monomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, then σ(K)=K.


(1)(2) Let X be an F-basis for K, and for each xX, let fx be the irreducible polynomialMathworldPlanetmath of x over F. By hypothesisMathworldPlanetmath, each fx splits over K, and because we evidently have K=F(X), it follows that K is a splitting field of {fx:xX} over F.
(2)(3) Assume that K is a splitting field over F of SF[X]. Given fS, we may write f(X)=ui=1n(X-ui) for some u,u1,,unK; because σ fixes F pointwise, we have σ(ui){u1,,un} for 1in, and since σ is injectivePlanetmathPlanetmath, it must simply permute the roots of f. Thus u1,,unσ(K). As K is generated over F by the roots of the polynomials in S, we obtain K=σ(K).
(3)(1) Let K¯ be an algebraic closure of K, noting that, since K is algebraic over F, that same is true of K¯, and consequently K¯ is an algebraic closure of F containing K. Now suppose fF[X] is irreduciblePlanetmathPlanetmath and that uK is a root of f, and let v be any root of F in K¯. There exists an F-isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath τ:F(u)F(v) such that τ(u)=v. Because K¯ is a splitting field over both F(u) and F(v) of the set of irreducible polynomials in F[X], τ extends to an F-isomorphism σ:K¯K¯. It follows that σ|K:KK¯ is an F-monomorphism, so that, by hypothesis, σ(K)=K, hence that v=σ(u)K. Thus f splits over K, and therefore K/F is normal. ∎

Title equivalent conditions for normalityPlanetmathPlanetmath of a field extension
Canonical name EquivalentConditionsForNormalityOfAFieldExtension
Date of creation 2013-03-22 18:37:56
Last modified on 2013-03-22 18:37:56
Owner azdbacks4234 (14155)
Last modified by azdbacks4234 (14155)
Numerical id 8
Author azdbacks4234 (14155)
Entry type Theorem
Classification msc 12F10
Related topic NormalExtension
Related topic SplittingField
Related topic ExtensionField
Related topic AlgebraicExtension