A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
A Brief Introduction to Classical and Adelic Algebraic ... - William Stein
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INDEX 189<br />
restricted <strong>to</strong>pological product, 157<br />
ring<br />
of N-adic numbers, 122<br />
ring of integers, 26, 111<br />
Salem number, 177<br />
same <strong>to</strong>pology implies equivalent valuations<br />
proposition, 118<br />
Selmer, 125<br />
Selmer curve, 125<br />
separable, 140<br />
Shafarevich-Tate group, 125<br />
short exact sequence, 19<br />
Smith normal form, 15<br />
Smith normal form proposition, 16<br />
strong approximation theorem, 165<br />
structure of p n /p n+1 proposition, 60<br />
structure of abelian groups theorem,<br />
15<br />
structure of tensor product of fields<br />
lemma, 140<br />
structure theorem, 15<br />
subgroup of free group proposition, 15<br />
submodule, 19<br />
subset <strong>to</strong>pology on 1-ideles I 1 K lemma,<br />
169<br />
surjection <strong>and</strong> Noetherian lemma, 21<br />
symmetric about the origin, 68<br />
tensor product <strong>to</strong>pology, 140<br />
tensor products <strong>and</strong> characteristic polynomials<br />
corollary, 142<br />
Theorem<br />
chinese remainder, 58<br />
compact quotient of adeles, 160<br />
compact quotient of ideles, 170<br />
compactness of ring of integers, 131<br />
complete embedding, 119<br />
Dirichlet unit, 77<br />
Faltings, 12<br />
finiteness of class group, 67, 171<br />
finiteness of function field class group,<br />
172<br />
fractional ideals, 33<br />
Gelf<strong>and</strong>-Tornheim, 113<br />
Hasse-Minkowski, 125<br />
Hilbert Basis, 21<br />
Ostrowski, 113<br />
prime ideal fac<strong>to</strong>rization, 53<br />
product formula, 154<br />
product of extensions, 152<br />
properties of Haar measure, 133<br />
reduction of Galois group, 99<br />
strong approximation, 165<br />
structure of abelian groups, 15<br />
transitive Galois action, 93<br />
unique ideal fac<strong>to</strong>rization, 35<br />
uniqueness of valuation extension,<br />
145<br />
valuation extensions, 149<br />
valuations on Q, 113<br />
weak approximation, 125<br />
<strong>to</strong>pological field, 117<br />
<strong>to</strong>pological field lemma, 117<br />
<strong>to</strong>pology on adeles corollary, 157<br />
<strong>to</strong>tally disconnected, 124<br />
trace, 27<br />
trace pairing, 64<br />
trace pairing nondegenerate lemma, 65<br />
transitive Galois action theorem, 93<br />
triangle inequality proposition, 108<br />
trivial valuation, 107<br />
unique ideal fac<strong>to</strong>rization theorem, 35<br />
uniqueness of valuation extension theorem,<br />
145<br />
unit norm characterization proposition,<br />
78<br />
valuation, 107, 129<br />
discrete, 109<br />
equivalence of, 107<br />
valuation extensions theorem, 149<br />
valuation stays non-archimedean corollary,<br />
120<br />
valuations on Q theorem, 113<br />
valuations on k(t) lemma, 116