Algebraic Number Theory

a e F a₁ algebraic number field archimedean primes class number completes the proof Consider Corollary DE/F Dedekind denote discrete prime divisor discriminant divisor of F exists extension E/F extension of F fact factorization field F finite extension finite number follows Furthermore Galois extension global field homomorphism ideal class ideal of F idèle integral basis integral closure integral ideal irreducible isomorphism lattice Lemma Let F mod Q Moreover multiplicative nontrivial norm notation polynomial prime ideal principal ideal domain product formula proper prime ideal Proposition prove PS(F quadratic field quotient field r₁ residue class residue class field restricted direct product ring of integers roots of unity subgroup subset Suppose that F topology totally ramified trivial unique extension unramified vp(a vp(b ZQ/P