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
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matching integers lemma
metric
minimal polynomial
minimal polynomial of algebraic integer lemma
Minkowski
multiplicativity of ideal norm proposition
[email protected]$ N$-adic
numbers
[email protected]$ 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


William Stein 2004-05-06