All Questions
Tagged with p-adic-numbers reference-request
9 questions with no upvoted or accepted answers
11
votes
0
answers
428
views
Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
- 631
8
votes
0
answers
585
views
Scholze and Weinstein's $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}_p$
In their Berkeley Lectures, to motivate the introduction of Diamonds, Scholze and Weinstein discuss what should be the definition of $\operatorname{Spa}\mathbf{Z}_p\times \operatorname{Spa}\mathbf{Z}...
- 1,479
6
votes
0
answers
698
views
Jacobson radical of a tensor product
Let $R$ be a commutative ring and $A_1$, $A_2$ be $R$-algebras. Is there any general mean to compute the Jacobson radical of the tensor product $A_1\otimes_R A_2$ in terms of ${\rm Rad}(A_i )$, $i=1,2$...
- 6,349
4
votes
0
answers
307
views
Computing the $2$-adic volume of a special orthogonal group
Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\...
3
votes
0
answers
253
views
Jacobian change of variables formula for $p$-adic valued integration?
Let $k$ be a $p$-adic field. It's possible to make sense of the Haar measure $\mu_{\operatorname{Haar}}$ on $k^n$ as a $k$-valued measure and define integrals
$$\int\limits_{k^n} f(x_1, ... , x_n) d\...
- 6,180
3
votes
0
answers
164
views
Using the Hilbert symbol to find nice field extensions
Let $p$ and $q$ be (not necessarily distinct) odd primes and let $F=\mathbb{Q}_p(\mu_q)$. The $q^{th}$ Hilbert symbol induces a non-degenerate alternating form $$(\cdot,\cdot)_q:F^\times/(F^\times)^q\...
- 2,516
2
votes
0
answers
172
views
Bruhat-Tits tree as Cayley graph of free group
$\DeclareMathOperator\BT{BT}\DeclareMathOperator\GL{GL}$Let $p > 2$ be a prime and $n = \frac{p + 1}{2}$. We can identify the vertices of Bruhat-Tits tree $\BT(\mathbb Q_p)$ with the elements in ...
- 71
2
votes
0
answers
266
views
A "multi-adic" absolute value / topology?
Let $S$ be a set of finitely many prime numbers. Then, define $\left|\cdot\right|_{S}:\mathbb{Q}\rightarrow\left[0,\infty\right)$ by: $$\left|x\right|_{S}\overset{\textrm{def}}{=}\prod_{p\in S}\left|x\...
- 1,284
1
vote
0
answers
330
views
Automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers?
I am interested to see what is currently known about the automorphisms of the topological field $\mathbb{C}_p$ of $p$-adic complex numbers (with respect to the $p$-adic topology induced by the $p$-...
