# Tag Info

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### Has any one seen this sum of roots of unity before?

Using the sagemath code, ...

### 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 ...
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### 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 ...

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