1
vote
0answers
26 views

Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
5
votes
2answers
68 views

Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
3
votes
0answers
29 views

GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
1
vote
1answer
33 views

Writing $p$-groups using $p$-adics

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.
2
votes
1answer
114 views

Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$?

Let $p$ be an odd prime number and $n$ any positive integer. Is $(\mathbb Z/p^{n+1}\mathbb Z)^\times\cong (\mathbb Z/p\mathbb Z)^\times\times(\mathbb Z/p^n\mathbb Z)$ as groups? This seems very ...
3
votes
2answers
129 views

Where do $p$-adic numbers and $p$-Sylow theory both appear?

Both $p$-adic numbers and $p$-Sylow theory are by design "arithmetic" ways of "localizing," so it stands to reason they might be in cahoots in certain contexts. Are they?
5
votes
0answers
56 views

Why are the p-adic integers a linearly ordered group? [duplicate]

In a previous question, someone suggested the p-adic integers as an example of a non-archimedean linearly ordered group. I'm not sure why these are linearly ordered - specifically, it doesn't seem to ...
4
votes
3answers
338 views

Why multiplicative group $\mathbb{Z}_n^*$ is not cyclic for $n = 2^k$ and $k \ge 3$

Let G be the multiplicative group $\mathbb{Z}_n^*$ for $n = 2^k$ and $k \ge 3$. Can we prove that no element has order bigger than $2^{k-2}$ ? My solution (not really a solution) : Since $n=2^k$, I ...
2
votes
2answers
447 views

The $p$-adic integers as a profinite group

How to prove that if $\mathbb{Z}_p$ is the set of $p$-adic integers then $\displaystyle{\mathbb{Z}_p=\varprojlim\mathbb{Z}/p^n\mathbb{Z}}$ where the limit denotes the inverse limit? $\mathbb{Z}_p$ is ...
2
votes
1answer
78 views

normal groups of a infinite product of groups

I have a question regarding the quotient of a infinite product of groups. Suppose $(G_{i})_{i \in I}$ are abelian groups with $|I|$ infinite and each $G_i$ has a normal subgroup $N_i$. Is it true in ...
3
votes
1answer
232 views

Weird quotient of $\langle\mathbb Q,+\rangle$?

after looking at this question I came to think on one particular case. I'm wondering if maybe I've missed something on the way. If anyone could give it a look that would be great: We start by ...