3
votes
0answers
36 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
3
votes
1answer
145 views

Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?

If $p$ and $q$ are distinct prime number, are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic as topological space?
3
votes
1answer
78 views

How can every $p$-adic integer be the limit of a sequence of non-negative integers?

See Andrew Baker's P-adic Notes. Every element of $\mathbb{Z}_p = \{a \in \mathbb{Q}_p : |a|_p \leq 1 \}$ is a limit of a sequence of non-negative integers, with respect to the $|\cdot|_p$ norm. How ...
3
votes
1answer
109 views

Reference requested: 'decomposition' of Haar measure on the adeles.

Since the adeles $\mathbb{A}$ (with addition) are a locally compact Hausdorff topological group there exists a Haar measure $\mu$. Now people claim that it can be normalized such that for every ...
5
votes
2answers
776 views

Why are closed balls in the $p$-adic topology compact?

I was skimming through some of this paper Measurable Dynamics and Simple $p$-adic Polynomials out of curiosity. A few pages in, the author claims that closed balls are both open and compact sets in ...