Tagged Questions
2
votes
2answers
35 views
Valuations, Isomorphism, Local ring
Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$.
Let $v_p(x):=r$ and $v_p(0):= \infty$. Also,
$$\mathcal O_p= \left\{ ...
1
vote
1answer
64 views
Valuation rings of complete non-archimedean fields which are not local
I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
3
votes
1answer
87 views
A property of non-Archimedean metrics
I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers,
$p$-adic Analysis, and Zeta-Functions), and came across the exercise:
Prove that a norm $\|.\|$ on ...
9
votes
4answers
265 views
Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?
I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this:
Show that $x^2-82y^2=\pm2$ has solutions in every ...
6
votes
1answer
162 views
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$.
I was trying ...
4
votes
2answers
403 views
Roots of unity in a local field
The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as
$K=\langle \pi\rangle\times \mu_{q-1}\times ...
