# Tag Info

Accepted

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

Using the sagemath code, ...
• 56

### 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 ...
• 123
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 ...
• 4,129

• 21
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.
• 473
Accepted

• 1,349
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 $L$ ...
• 115k
### 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_\...