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1-1 mapping 3-cycles a e G abelian group algebraic over F automorphism commutative ring complex numbers conjugate constructible Corollary cyclic group defined definition degree Eisenstein Criterion element of G equivalence relation Example extension of F finite extension finite field finite group given greatest common divisor group G group of order H in G hence homomorphism homomorphism of G Homomorphism Theorem ideal of F[x induction integral domain inverse irreducible in Q[x isomorphic kernel left cosets Lemma Let F Let G maximal ideal Middle-Level Problems minimal polynomial monic monic polynomial nonabelian group nonzero elements normal subgroup number of elements permutation positive integer prime number proves the theorem rational numbers reader real numbers relatively prime result of Problem Section space over F subgroup of G subgroup of order subset Suppose Sylow subgroup transpositions unique unit element vector space verify x e G Zp[x