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Outline of Galois Theory Development - Stanford University

Outline of Galois Theory Development1. Field extensionF Eas vector space overF.|E:F|equals dimension as vector space. IfF K Ethen|E:F|=|E:K||K:F|.2. Elementa Eis algebraic overFif and only if|F(a) :F|is finite. Minimum polynomialf(X) F[X]for algebraica (X) is irreducible,F(a) =F[a] =F[X]/(f(X)),|F[a] :F|= degf(X), a basisis{1,a,a2,..,ad 1}, whered= degf(X). The set of elementsa Ewhich are algebraic overFis asubfield Existence of Splitting Fields for a polynomial or family of polynomialsF F[X]. Existence ofAlgebraic Closure. Characterizations of Algebraic Closure: A fieldEis algebraically closed if everynon-constant polynomial inE[X] factors as a product of linear polynomials. (Equivalently, every non-constant polynomial inE[X] has a root inE.)

10. De ne E=Fto be a Galois extension if and only if Eis separable AND normal over F. (This is the ’right’ de nition, because the conditions separable and normal are easily understood in terms of individual

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