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College Algebra Exam Review 440

College Algebra Exam Review 440 - 450 9 FIELD EXTENSIONS...

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450 9. FIELD EXTENSIONS – SECOND LOOK Proposition 9.7.2. Let f 2 KOExŁ be an irreducible separable polynomial. Let E be a splitting field for f , and let ˛ 1 ; : : : ; ˛ n be the roots of f .x/ in E . The Galois group Aut K .E/ , regarded as a group of permutations of the set of roots, is contained in the alternating group A n if, and only if, the discriminant ı 2 .f / of f has a square root in K . Proof. The discriminant has a square root in K if, and only if, ı.f / 2 K . But ı.f / 2 K if, and only if, it is fixed by the Galois group, if, and only if, the Galois group consists of even permutations. n We do not need to know the roots of f in order to compute the dis- criminant. Because the discriminant of f D P i a i x i is the discriminant of the monic polynomial .1=a n /f multiplied by a 2n 1 n , it suffices to give a method for computing the discriminant of a monic polynomial. Suppose, then, that f is monic. The discriminant is a symmetric polynomial in the roots and, there- fore, a polynomial in the coefficients of f , by Theorem 9.6.6
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