**-module****span and lemma****is a lattice proposition****is Noetherian corollary****is integrally closed proposition****is Dedekind proposition****divides product of primes lemma****, , proposition****lemma****-adic valuation****-adic distance****-distance is metric proposition****-adic numbers****-adic field****and are totally disconnected lemma****and base extension corollary****-ideles****lemma****totally disconnected proposition****is a PID proposition****is a ring proposition****has finite measure corollary****lemma****lemma****abelian groups****structure theorem**

**adele ring****adic-expansion lemma****algebraic integer****Algebraic number theory****almost all****any two norms equivalent lemma****archimedean****Artin symbol****ascending chain condition****base extension of adeles lemma****Birch and Swinnerton-Dyer conjecture****Blichfeld lemma****Blichfeldt's lemma****Cauchy sequence****characterization of discrete lemma****characterization of integrality proposition****characterization of Noetherian proposition****chinese remainder theorem****class group**,**class group generated by bounded primes lemma****cokernel****compact quotient of adeles theorem****compact quotient of ideles theorem****compact subset of adeles corollary****compactness of ring of integers theorem****complete**,**complete embedding theorem****complete local field locally compact corollary****complete|nn****completion****completion, norms, and traces corollary****complex -dimensional representation****conjugation of Frobenius proposition****connected****connected|nn****content****content map is continuous lemma****convex****Corollary****is Noetherian****and base extension****has finite measure****compact subset of adeles****complete local field locally compact****completion, norms, and traces****discrimant of number field****extension of complete field is complete****factorization of fractoinal ideals****group as quotient of free groups****norm, trace compatible with towers****norms, traces, and completions****order of inertia group****tensor products and characteristic polynomials****topology on adeles****valuation stays non-archimedean****value set stays same**

**cyclotomic fields****decomposition group****decomposition groups are conjugate lemma****Dedekind domain****dimension of embedding of field proposition****Dirichlet unit theorem****disconnected****discrete****discrete subgroup of proposition****discrimant of number field corollary****discriminant**, ,**discriminant of order proposition****divides****elliptic curve****equivalent**,**equivalent non-archimedean valuations and 's lemma****equivalent valuations, same topology lemma****essential discriminant divisor****Euler****exactness and Noetherian lemma****extends****extension of complete field is complete corollary****extension of normalized valuation lemma****factorization of fractoinal ideals corollary****factorization of lemma****Faltings theorem****Fermat's last theorem****field of fractions****finitely generated****finiteness of class group theorem**,**finiteness of function field class group theorem****fixed field characterization proposition****fractional ideal****fractional ideal is lattice lemma****fractional ideals and formal sums of valuations lemma****fractional ideals theorem****Frobenius element****Galois**,**Galois conjugates****Gelfand-Tornheim theorem****global field****group as quotient of free groups corollary****group of units****Haar measure****Haar measure on compact lemma****Haar measure on lemma****Hasse****Hasse-Minkowski theorem****Hensel's lemma****Hilbert Basis theorem****homomorphism****hyperplane embedding lemma****icosahedral****ideals generated by two elements proposition****idele group****ideles are a restricted product lemma****inertia group****inertia group characterization proposition****inertia subgroup****integral ideal****integral ideals of bounded norm lemma****integrally closed in its field of fractions****lattice index****lattices and volumes lemma****Lemma****span and****divides product of primes****and are totally disconnected****adic-expansion****any two norms equivalent****base extension of adeles****Blichfeld****characterization of discrete****class group generated by bounded primes****content map is continuous****decomposition groups are conjugate****equivalent non-archimedean valuations and 's****equivalent valuations, same topology****exactness and Noetherian****extension of normalized valuation****factorization of****fractional ideal is lattice****fractional ideals and formal sums of valuations****Haar measure on compact****Haar measure on****hyperplane embedding****ideles are a restricted product****integral ideals of bounded norm****lattices and volumes****local compactness of restricted product****matching integers****minimal polynomial of algebraic integer****non-archimedean valuation characterization****open ball is closed****principal ideles are discrete****reduction homomorphism****restricted product****structure of tensor product of fields****subset topology on -ideles****surjection and Noetherian****topological field****trace pairing nondegenerate****valuations on****valuations such that****volume of rings of integers**

**lies over****local compactness of restricted product lemma****local-to-global principal****locally compact****MAGMA**, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,**matching integers lemma****metric****minimal polynomial****minimal polynomial of algebraic integer lemma****Minkowski****multiplicativity of ideal norm proposition****N@-adic****numbers**

**N@-adic****totally disconnected**

**nilpotent****Noetherian**,**Noetherian equals finitely generated proposition****non-archimedean****non-archimedean valuation characterization lemma****nontrivial solution****norm**, ,**norm and trace proposition****norm, trace compatible with towers corollary****normalized****norms, traces, and completions corollary****number field**,**open ball is closed lemma****open balls****open problem****solvability of plane cubics**

**order**,**order of inertia group corollary****Ostrowski theorem****prime ideal factorization theorem****principal adeles****principal ideles****principal ideles are discrete lemma****product formula theorem****product measure****product of extensions theorem****properties of Haar measure theorem****Proposition****is a lattice****is integrally closed****is Dedekind****, ,****-distance is metric****totally disconnected****is a PID****is a ring****characterization of integrality****characterization of Noetherian****conjugation of Frobenius****dimension of embedding of field****discrete subgroup of****discriminant of order****fixed field characterization****ideals generated by two elements****inertia group characterization****multiplicativity of ideal norm****Noetherian equals finitely generated****norm and trace****same topology implies equivalent valuations****Smith normal form****structure of****subgroup of free group****triangle inequality****unit norm characterization**

**radical****reduction homomorphism lemma****reduction of Galois group theorem****residue class degree****restricted product lemma****restricted topological product****ring of integers**,**ring****of -adic numbers|nn**

**Salem number****same topology implies equivalent valuations proposition****Selmer****Selmer curve****separable****Shafarevich-Tate group****short exact sequence****Smith normal form****Smith normal form proposition****strong approximation theorem****structure of abelian groups theorem****structure of tensor product of fields lemma****structure of proposition****structure theorem****subgroup of free group proposition****submodule****subset topology on -ideles lemma****surjection and Noetherian lemma****symmetric about the origin****tensor product topology****tensor products and characteristic polynomials corollary****Theorem****chinese remainder****compact quotient of adeles****compact quotient of ideles****compactness of ring of integers****complete embedding****Dirichlet unit****Faltings****finiteness of class group**,**finiteness of function field class group****fractional ideals****Gelfand-Tornheim****Hasse-Minkowski****Hilbert Basis****Ostrowski****prime ideal factorization****product formula****product of extensions****properties of Haar measure****reduction of Galois group****strong approximation****structure of abelian groups****transitive Galois action****unique ideal factorization****uniqueness of valuation extension****valuation extensions****valuations on****weak approximation**

**topological field****topological field lemma****topology on adeles corollary****totally disconnected**,**trace****trace pairing****trace pairing nondegenerate lemma****transitive Galois action theorem****triangle inequality proposition****trivial valuation****unique ideal factorization theorem****uniqueness of valuation extension theorem****unit norm characterization proposition****valuation**,**valuation extensions theorem****valuation stays non-archimedean corollary****valuation****discrete****equivalence of**

**valuations on lemma****valuations on theorem****valuations such that lemma****value set stays same corollary****volume****volume of rings of integers lemma****weak approximation theorem**

William Stein 2004-05-06