The p-adic-number-theory tag has no wiki summary.
3
votes
0answers
273 views
p-adic numbers and binomial coefficients
Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$
$${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$
This is again a ...
4
votes
0answers
127 views
Is there a notion of *p-adic Dedekind Domains*?
As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$.
Now is there any generalization such as the p-adic completions of a Dedekind Domain?
This might be ...
6
votes
2answers
146 views
Does $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ depend on the embedding of $\overline{\mathbb{Q}}$ into $\overline{\mathbb{Q}_p}$?
I think the title pretty much says it all. I'm getting confused in the subtle parts of a proof, and would appreciate some help.
8
votes
2answers
437 views
Are all nonarchimedean valuations discrete?
I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. ...
6
votes
2answers
352 views
$n$th powers in the p-adics
Suppose $K$ is a $p$-adic field (finite extension of the $p$-adics), and let $n$ be any integer (independent of what $p$ is). Define $U$ to be the set of all $x$ in $K$ such that $|x| = 1$ and such ...
16
votes
3answers
973 views
How far are the $p$-adic numbers from being algebraically closed?
A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as ...
1
vote
2answers
265 views
Computing Newton polygons
I am looking at the first two examples in Paul Garretts notes on p-adic number theory.
The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower ...
6
votes
2answers
510 views
Binomial coefficients: how to prove an inequality on the $p$-adic valuation?
In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used:
$$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...
