## New answers tagged algebraic-number-theory

4
votes

### Class numbers in the unramified biquadratic extensions of number fields

Kuroda's class number formula does not require the extension to be unramified.
It is not true, as you write, that the class number of $K$ can be determined from the class numbers of the $k_i$; you ...

4
votes

### Equidistribution on $\mathrm{SU}_2$

It was asked in the comments that I provide some details. I prove slightly more: if $\mu_n$ denotes uniform probability on the sphere of radius $n$ and if $\rho:F_2 \to \mathrm{SU}_2$ is a ...

8
votes

Accepted

### Equidistribution on $\mathrm{SU}_2$

In the article "On the spectral gap for finitely-generated subgroups of SU(2)" by Jean Bourgain and Alex Gamburd (Invent. Math. 171, No. 1, 83-121 (2008)), they show that free subgroups of $...

5
votes

Accepted

### Generators of the ideal class group

Let $G$ be the ideal class group, and let $H$ be the subgroup generated by the prime ideals of norm at most $3\log^2(d^2)=12\log^2(d)$. Assume that $H$ is a proper subgroup of $G$. Then there is a ...

2
votes

Accepted

### How to compute the asymptotic constant for the count of $S_3$-sextic number fields?

I will do a worked example for the tame case and show that it agrees with the formula from their paper. You should then be able to adapt this to the case $p=2,3$.
Let $p > 3$. I use formula (7) ...

5
votes

Accepted

### Ramification criteria for Kummer extensions

$K( A^{1/n})$ is unramified if and only if the image of $A$ in $K_{\mathfrak p}^* / (K_{\mathfrak p}^*)^n$ lies in a certain cyclic subgroup of order $n$, which is the cyclic subgroup generated by a ...

3
votes

Accepted

### Conductor and local Kroneckerâ€“Weber theorem

Let $m$ be the order of $p \in \mathbb Q_p^\times$ in the character of $\mathbb Q_p^\times$ associated to $F$ by local class field theory.
Let $\ell$ be the least integer prime to $p$ such that the ...

0
votes

### Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$

I added this non-answer since it is too long for a comment.
EDIT: In the first version I claimed that the action is free, which it is not.
There is a very explicit model for the classifying space $...

Community wiki

5
votes

Accepted

### Cohomology of $S$-arithmetic groups with trivial coefficients such as $H^n(\rm{PGL}_2(\mathbb{Z}[1/N]);\mathbb{Z})$

This is a hard problem in general. When $N$ is not prime, then even the first homology of $\operatorname{PSL}_2(\mathbf{Z}[\frac{1}{N}])$ is non-trivial to compute (cf. Corollary 4.4 of [1]), but I ...

1
vote

### Is the completed tensor product (over a complete dvr) of two reduced complete Noetherian local rings again reduced?

Let $K$ be the fraction field of $\mathcal{O}$.
Assuming $A$ and $B$ are $\mathcal{O}$-flat, then also $A \widehat{\otimes} B$ is $\mathcal{O}$-flat (exercise), so it's enough to see that $C:=(A \...

3
votes

### Explicit description of the extension generated by the square root of a fundamental unit of a real quadratic field

This has nothing to do with ramification. Write your unit in the form $\varepsilon = t + u \sqrt{m}$; then $t^2 - mu^{2} = 1$, hence $t^2 - 1 = (t-1)(t+1) = mu^2$. If $t$ is odd, then unique ...

Top 50 recent answers are included

#### Related Tags

algebraic-number-theory × 2143nt.number-theory × 1316

ag.algebraic-geometry × 290

class-field-theory × 200

reference-request × 180

analytic-number-theory × 177

galois-theory × 165

arithmetic-geometry × 163

number-fields × 138

ac.commutative-algebra × 129

elliptic-curves × 128

modular-forms × 88

local-fields × 69

galois-representations × 68

polynomials × 65

gr.group-theory × 59

prime-numbers × 56

cyclotomic-fields × 55

computational-number-theory × 53

algebraic-groups × 52

rt.representation-theory × 50

diophantine-equations × 47

ra.rings-and-algebras × 45

fields × 45

iwasawa-theory × 44