Algebraic Number Theory Introduction

Galois group, 222adeles, 149Artin’s conjecture, 293automorphic form, 292bar resolution, 172biquadratic reciprocity, 271Brauer group, 205Brauer-Hasse-Noether theorem, 252central simple algebra, 206centralizer, 206chain map, 163change of group, 184Chebotarev density theorem, 282class formation, 222class group, 33cohomological functor, 169cohomology, 164cohomology of lattices, 245cohomology of units, 214coinduced module, 175Comparison theorem, 164complete resolution, 178complex, 163complex multiplication, 301conductor, 148congruence subgroup, 146corestriction, 184cubic reciprocity, 274cup product, 181cusp form, 296Cyclotomic polynomials, 71decomposition group, 26Dedekind domain, 18degree equation, 22density, 282derivation, 173derived functors, 165descent, 202dimension shifting, 180Dirichlet’s S-unit theorem, 69Dirichlet’s theorem for number fields, 283Dirichlet’s unit theorem, 65discrete valuations, 17double centralizer theorem, 206existence theorem, 235Ext, 165factor set, 173first inequality, 242formation, 223Frobenius element, 139fundamental unit, 224Galois cohomology, 197Galois representation, 293Gauss composition, 58global reciprocity, 147Grössencharacter, 294Grössencharacter of elliptic curve, 322group cohomology, 170group homology, 174group ring, 170Hasse norm theorem, 276Hasse-Minkowski, 277Hecke character, 294Hensel’s lemma, 93Herbrand quotient, 192Hilbert class field, 288Hilbert class field of imaginary quadratic field,310Hilbert symbol, 232