Unformatted text preview: ± n .x i ± x j /: We know that ı distinguishes even and odd permutations. Every permutation ± satisﬁes ±.ı/ D ˙ ı and ± is even if, and only if, ±.ı/ D ı . The symmetric polynomial ı 2 is called the discriminant polynomial . Now let f D P i a i x i 2 KŒxŁ be polynomial of degree n . Let ˛ 1 ;:::;˛ n be the roots of f.x/ in a splitting ﬁeld E . The element ı 2 .f / D a 2n ² 2 n ı 2 .˛ 1 ;:::;˛ n / is called the discriminant of f . Now suppose in addition that f is irreducible and separable. Since ı 2 .f / is invariant under the Galois group of f , it follows that ı 2 .f / is an element of the ground ﬁeld K . Since the roots of f are distinct, the element ı.f / D a n ² 1 n ı.˛ 1 ;:::;˛ n / is nonzero, and an element ± 2 Aut K .E/ induces an even permutation of the roots of f if, and only if, ±.ı.f // D ı.f / . Thus we have the following result:...
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Galois theory, ring isomorphism, j Än

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