Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with algebraic-number-theory
Search options not deleted
user 95241
Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
9
votes
1
answer
550
views
Absolute values of non-unit algebraic integers at all infinite places
Let $K$ be a number field and $O_K$ its ring of integers. Given any non-unit element $\alpha\in O_K$, does there exist a unit $u\in O^\times_K$ such that
$$
|\sigma(u \alpha)| >1 \quad \text{ for all …
- 1,053
2
votes
1
answer
438
views
class field theory for K-groups of number fields
Let $F$ be a number field. The classical class field theory gives an isomorphism between the ideal class group $Cl(F)$ and $\mathrm{Gal}(H/F)$, where $H$ is the Hilbert class field (maximal unramifie …
- 1,053
9
votes
Are quadratic units cyclotomic norms?
Edit:Let $K=\mathbb Q(\zeta_p)^+$
Since it's easy to show $-1 \in N(\mathcal O_K ^{\times})$ and $\mathrm{Gal}(K/k)$ is cyclic, your question is equivalent to ask whether $\mathrm{H}^2(K/k,\mathcal O …
- 1,053
1
vote
Generalized Greenberg's conjecture for imaginary quadratic fields
Let $K=\mathbb{Q}(\sqrt{-q})$ where $q\equiv 7\bmod 16$ is a prime. Take $F=K(\sqrt[4]{-q},\sqrt{-1})$. Then we proved that $X_{F\tilde{K}}=0$; see the paper (https://link.springer.com/article/10.100 …
- 1,053
14
votes
1
answer
1k
views
How Dirichlet proved Dirichlet's unit theorem for general number fields?
For a general number field $K$, Dirichlet's unit theorem states that the unit group of the ring of integers of $K$ is a finitely generated group of rank $r_1+r_2-1$.
It seems that standard algebraic n …
- 1,053
3
votes
1
answer
277
views
What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?
If $K/\mathbb{Q}$ is an infinite algebraic extension, define as usual the class group $Cl_K$ by the direct limit via the natural (conorm) map $Cl_K := \lim\limits_{\rightarrow} Cl_F$,
where $F$ runs o …
- 1,053
9
votes
1
answer
513
views
class number of prime degree field with prime conductor
Let $K$ be an finite abelian extension of $\mathbf{Q}$ conductor $p$, where $p$ is an odd prime. That is, $K \subset \mathbf{Q}(\mu_ p)$, the $p$-th cyclotomic field. Let $h_K$ be the class number of …
- 1,053
4
votes
0
answers
390
views
Is every Dedekind domain the integral closure of some principal ideal domain?
I mean that $B$ is a Dedekind domain with fraction field $L$, which is a finite separable extension of a field $K$ that is the fraction field of a PID $A$ such that $B$ is the integral closure of $A$ …
- 1,053