All Questions
Tagged with p-adic-numbers gr.group-theory
11 questions
4
votes
1
answer
585
views
Subgroup of p-adic units
Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
- 492
1
vote
1
answer
89
views
Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
- 21.8k
-1
votes
1
answer
300
views
Is this submonoid of the isometry group on $\Bbb Q_2$ closed to inverses? [closed]
Let $\textrm{aff}(ax+b)$ be the affine group on $\Bbb Z_2^\times$
i.e. the set of linear polynomials over 2-adic numbers with $a\in\Bbb Z_2^\times, b\in\Bbb Z_2$
Now let $X$ be the restriction of its ...
- 308
7
votes
1
answer
348
views
Topological generators for $\mathrm{SL}_2(\mathbf{Z}_p)$
$\DeclareMathOperator\SL{SL}$ Let $p>3$ and $G$ be an open subgroup of the special linear group $\SL_2(\mathbf{Z}_p)$ over the ring $\mathbf{Z}_p$ of $p$-adic integers. Suppose that $G$ is ...
4
votes
1
answer
243
views
Existence of intermediate field extensions for tamely ramified p-adic extensions
Let $p$ be a prime, and let $K/\mathbb{Q}_p$ be a tamely ramified finite extension of degree $n$. Let $q$ be a prime factor of $n$ with $q\neq p$. Must there exist an intermediate extension $L$ (...
- 421
1
vote
1
answer
428
views
Characters of p-adic units
Let $p$ be a prime and denote by $\mathbb{Z}_p^{\times}$ the group of $p$-adic units. Suppose that $\chi$ is a character $\chi: \mathbb{Z}_p^{\times} \rightarrow \mathbb{C}^{\times}$. Then it is well ...
- 59
3
votes
0
answers
94
views
Projective limit of copies of same group w.r.t. some fixed endomorphism
In our study of automorphism groups of transcendental field extensions, we have encountered the situation where we have a group $F$ together with an endomorphism $\alpha \colon F \to F$, resulting in ...
- 6,614
7
votes
0
answers
489
views
intuition for lattices in p-adic (or other non-Archimedean) vector spaces?
I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$.
I have some intuition for $\mathbb{Z}$-lattices ...
- 1,338
2
votes
1
answer
413
views
$2$-adic valuation on $\mathbb Q (\sqrt{-15})$
I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ ...
- 243
2
votes
1
answer
695
views
The Unit Group of $\mathbb{Z}_p$
Let $\mathbb{Z}_p$ the ring of $p$-adic numbers. It's known that the multiplicative unit group $\mathbb{Z}_p ^\times$ can be set theoretically described as $\bigcup _{1 \le a \le p-1} a+ p\mathbb{Z}_p$...
- 5,966
6
votes
1
answer
522
views
When do two lattices have the same stabilizer in the diagonal torus?
This is moved from MSE, where I asked and didn't receive an answer (see https://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...
- 1,453