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Principal Ideal Domains and Unique Factorization Domains
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4-module a,b e Abelian extension algebraic extension algebraic integers algebraic number field assume Chapter class number coefficients compute congruence conjugates consider contained cyclic group decomposition Dedekind domain deduce defined denote Determine discr discriminant distinct prime dL\K equal exists an element extension of degree fact field of quotients fractional ideal Galois extension Galois group hence homomorphism inertial degree integral basis integrally closed invertible isomorphism kernel Lemma linear linearly independent mapping matrix maximal ideal minimal polynomial multiplicative group natural number Noetherian ring non-zero fractional ideals non-zero integral ideal non-zero prime ideal norm odd prime prime number principal ideal domain Proof prove quadratic residue quadratic residue modulo real number relatively prime residue classes modulo ring of algebraic ring of integers root of unity Show Similarly subgroup submodule subring theorem unique factorization domain unit unramified