4509. FIELD EXTENSIONS – SECOND LOOKProposition 9.7.2.Letf2KOExŁbe an irreducible separable polynomial.LetEbe a splitting field forf, and let˛1; : : : ; ˛nbe the roots off .x/inE. The Galois groupAutK.E/, regarded as a group of permutations ofthe set of roots, is contained in the alternating groupAnif, and only if, thediscriminantı2.f /offhas a square root inK.Proof.The discriminant has a square root inKif, and only if,ı.f /2K.Butı.f /2Kif, and only if, it is fixed by the Galois group, if, and onlyif, the Galois group consists of even permutations.nWe do not need to know the roots offin order to compute the dis-criminant. Because the discriminant offDPiaixiis the discriminantof the monic polynomial.1=an/fmultiplied bya2n1n, it suffices to givea method for computing the discriminant of a monic polynomial. Suppose,then, thatfis monic.The discriminant is a symmetric polynomial in the roots and, there-fore, a polynomial in the coefficients off, by Theorem9.6.6
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