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1-1 mapping 3-cycles algebraic over F automorphism commutative ring complex numbers construct Corollary cyclic group defined definition degree Eisenstein Criterion element of G equations equivalence relation Example extension of F finite extension finite field finite group G F[x given greatest common divisor group G group of order hence homomorphism homomorphism of G ideal of F[x induction integral domain inverse irreducible in Q[x isomorphic kernel Lemma Let F Let G maximal ideal Middle-Level Problems monic polynomial multiplication nonabelian nonabelian group nonempty nonnegative normal subgroup nth roots number of elements permutation polynomial f(x positive integer prime number Prove that G proves the theorem rational numbers reader real numbers relatively prime result of Problem right cosets Section space over F subgroup of G subset Suppose theory transpositions unique unit element vector space verify