Index

$R$-module

$\O _K$ span and $\O _K\cap \mathbf{Q}=\mathbf{Z}$ lemma

$\O _K$ is a lattice proposition

$\O _K$ is Noetherian corollary

$\O _K$ is integrally closed proposition

$\O _K$ is Dedekind proposition

$I$ divides product of primes lemma

$e$, $f$, $g$ proposition

$\vert a+b\vert=\vert a\vert$ lemma

$N$-adic valuation

$N$-adic distance

$N$-distance is metric proposition

$N$-adic numbers

$p$-adic field

$K^+$ and $K^*$ are totally disconnected lemma

$\AA _K^+$ and base extension corollary

$1$-ideles

$\Bigl\vert\vert a\vert - \vert b\vert\Bigr\vert \leq \left\vert a-b\right\vert$ lemma

$\mathbf{Q}_N$ totally disconnected proposition

$\mathbf{Z}$ is a PID proposition

$\overline{\mathbf{Z}}$ is a ring proposition

$\AA _K^+/K^+$ has finite measure corollary

$I\cap{}J = IJ$ lemma

$\Norm (a I)$ 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 $n$-dimensional representation

conjugation of Frobenius proposition

connected

connected|nn

content

content map is continuous lemma

convex

Corollary

$\O _K$ is Noetherian

$\AA _K^+$ and base extension

$\AA _K^+/K^+$ has finite measure

compact subset of adeles

complete local field locally compact

completion, norms, and traces

discrimant of number field $>1$

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 $\mathbf{R}$ proposition

discrimant of number field $>1$ corollary

discriminant , ,

discriminant of order proposition

divides

elliptic curve

equivalent ,

equivalent non-archimedean valuations and $\O$'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 $p\O _K$ 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 $K^*$ 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

$\O _K$ span and $\O _K\cap \mathbf{Q}=\mathbf{Z}$

$I$ divides product of primes

$\vert a+b\vert=\vert a\vert$

$K^+$ and $K^*$ are totally disconnected

$\Bigl\vert\vert a\vert - \vert b\vert\Bigr\vert \leq \left\vert a-b\right\vert$

$I\cap{}J = IJ$

$\Norm (a I)$

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 $\O$'s

equivalent valuations, same topology

exactness and Noetherian

extension of normalized valuation

factorization of $p\O _K$

fractional ideal is lattice

fractional ideals and formal sums of valuations

Haar measure on compact

Haar measure on $K^*$

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 $1$-ideles $\mathbb{I}_K^1$

surjection and Noetherian

topological field

trace pairing nondegenerate

valuations on $k(t)$

valuations such that $\left\vert a\right\vert>1$

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@$N$-adic

numbers

N@$N$-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

$\O _K$ is a lattice

$\O _K$ is integrally closed

$\O _K$ is Dedekind

$e$, $f$, $g$

$N$-distance is metric

$\mathbf{Q}_N$ totally disconnected

$\mathbf{Z}$ is a PID

$\overline{\mathbf{Z}}$ is a ring

characterization of integrality

characterization of Noetherian

conjugation of Frobenius

dimension of embedding of field

discrete subgroup of $\mathbf{R}$

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 $\mathfrak{p}^n/\mathfrak{p}^{n+1}$

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 $N$-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 $\mathfrak{p}^n/\mathfrak{p}^{n+1}$ proposition

structure theorem

subgroup of free group proposition

submodule

subset topology on $1$-ideles $\mathbb{I}_K^1$ 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 $\mathbf{Q}$

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 $k(t)$ lemma

valuations on $\mathbf{Q}$ theorem

valuations such that $\left\vert a\right\vert>1$ lemma

value set stays same corollary

volume

volume of rings of integers lemma

weak approximation theorem