## New answers tagged algebraic-number-theory

4
votes

Accepted

4
votes

### What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

To give an alternative answer, let us first recall the article "Construction of Locally Compact Near-Fields from $p$-Adic
Division Algebras" by Detlef Groger:
Fix a prime element $\pi_F$ of ...

3
votes

Accepted

### What is the quotient group $D^*/{F^*(1+P_D)}$ for a quaternion division algebra $D$ over a local field $F$?

Yes, we can.$\newcommand{\order}{\mathcal{O}}$ $\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\prim}{\mathcal{P}}$ $\newcommand{\F}{\mathbb{F}}$
First, let me remind you of the following explicit ...

25
votes

### Can we state the Riemann Hypothesis part of the Weil conjectures directly in terms of the count of points?

(1) We have $$ N_n(X) = \sum_{k = 0}^{2d} (-1)^k \mathrm{tr}\left(\mathrm{Frob}^n \colon H^k(X) \to H^k(X) \right) = \sum_{k = 0}^{2d} (-1)^k \sum_{i=1}^{ h^k(X)} \lambda_{k,i}^n$$
where $\lambda_{k,...

2
votes

Accepted

### On presentations of universal rings of deformations

Doesn't this kind of prove itself? Pick some elements $\alpha_1, \dots, \alpha_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t_i$ to $\alpha_i$ defines a ...

2
votes

### Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?

The question can be rephrased: Find the integers $n > 3$ for which there exist Gaussian integers $z$ such that
$z^n \pm z = \overline{z}^n \pm \overline{z}$. Rewriting the equation as $(z^n -\...

1
vote

### Completion reducing to localization on Noetherian rings

No. If $A$ is a complete noetherian local ring, then $A$ is complete for the adic topology defined by any other prime ideal.

12
votes

Accepted

### Rationality of field embeddings

The answer is yes. Suppose $S$ is nonempty. Write $K=\mathbb Q(\alpha)$ (using the primitive element theorem). Applying your assumption to $\alpha^n$ for all $n\in\mathbb N$ we get that all sums $\...

2
votes

### Cancellation of irreducibility for Galois conjugates

The answer to your new question is still no. I mentioned this problem (or rather, the group-theoretic reformulation given by Will Sawin) to my colleague Steve Humphries, and he found the following ...

8
votes

### Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?

Your conjecture is likely to be true. I was not able to check the details because I couldn't get all the references, but here are the main steps.
The main point is that if there is a Gaussian integer $...

8
votes

Accepted

### Cancellation of irreducibility for Galois conjugates

No.
Let $a, b \in \mathbb Q(i)$. Let $\alpha_1$ be a root of $x^3 + ax + b$. Let $L$ be the field generated by $i, \alpha_1, \overline{\alpha}_1$. Assume that $a,b$ are sufficiently general that $L$ ...

3
votes

Accepted

### What's the class group of $\mathbb{Q}^{\mathrm{ab}}$?

From Armand Brumer, The class group of all cyclotomic integers:
As an abelian group, $\mathrm{Pic}(O_\infty)$ is isomorphic to a countable direct sum of copies of $\mathbb Q/\mathbb Z$.
Here $O_\...

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