# Tag Info

Accepted

### Completion and algebraic closure

First you have to observe that since all extensions of the valuation to $\bar{K}$ are conjugate, $\hat{\bar{K}}$ is well-defined up to (non-unique) isomorphism. Now, since $\hat{\bar{K}}$ is complete ...
• 19.3k
Accepted

### Is it a valuation ring?

This is (essentially) a conjecture of Krull; a counter-example was given by P. Ribenboim: Sur une note de Nagata relative à un problème de Krull, Math. Zeit. 64, 159-168 (1956). Note that "...
• 37.3k
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### Extension of 2-adic valuation to the real numbers

No. The important thing to know is that, if $K \subseteq L$ is a field extension and $v: K \to \mathbb{R}$ is a valuation, then $v$ can be extended to $L$. So I can answer all of your questions by ...
• 151k
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### simple questions on topological rings arising in the context of Perfectoid Spaces

Firstly, I don't think you should start to learn about adic spaces by thinking about perfectoid spaces. The sensible examples of adic spaces for a beginner to think about are sane Noetherian things ...
• 40.8k
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### In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?

Zero is the only algebraic integer which has all its conjugates strictly inside the complex unit circle. (Look at the norm.) For explicit examples with conjugates on either side of the unit circle, ...
• 2,391

### Uniquely ordered commutative rings

In a field, an element can be made negative in some order iff it is not a sum of squares. Thus, $F$ is uniquely ordered iff for every $a\in F^\times$, $a$ or $-a$, but not both, is a sum of squares. ...
• 44.7k
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### Is every field the residue field of a discretely valued field of characteristic 0?

Yes, by Hasse-Schmidt ("Die Struktur diskret bewerteter Koerper", Crelle's Journal, 1934) for any field $k$ of characteristic $p$ there exists a strict Cohen ring $A$, which is a Noetherian, ...
• 8,837

### Ostrowski's Theorem for topological rings?

Thanks to YCor's examples in the comments, I decided this question was worth a deep dive. It turns out on the one hand that There are lots of exotic (Hausdorff) field topologies on $\mathbb Q$. ...
Accepted

### Subsequence of the cubes

Experimenting with a CAS suggests an induction. In order to handle the induction, we need to consider the forms of the numbers involved. $\frac{4^m-1}{3} = 1 + 2^2 + 2^4 + \cdots + 2^{2m-2}$ ...
• 6,571
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### Literature on non-Archimedean analogues of basic complex analysis results

Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information. ...
• 45.7k

### Is the integral closure of a valuation ring in a finite separable extension of its fraction field étale?

EDIT I may have overlooked the assumption that $L$ is contained in $\hat{K}$. In general, if $v$ is a valuation of $K$ and $L/K$ is finite separable then $L \otimes_K \hat{K}$ is reduced and thus ...
• 20.5k

### Ostrowski's Theorem for topological rings?

The following relevant classification result, due to Kowalsky and Dürbaum [2], appears in Appendix B of Engler and Prestel [1]. Let $(K,\tau)$ be a topological field. Then $\tau$ is called a V-...
• 44.7k

### Examples of NIP fields of characteristic $p$

For any field $K$ and ordered group $\Gamma$, the Malc'ev-Neumann field $K((\Gamma))$ is maximal by a Theorem of Krull, hence algebraically maximal. Taking a perfect infinite NIP field $K$, for ...
• 1,555
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### The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory

The Arakelov-theoretical analogue of Riemann's inequality is Minkowski's theorem. As Felipe Voloch's answer indicates, Tate's approach makes the Riemann-Roch theorem a consequence of the Poisson ...
• 12.8k
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### Open immersion of affinoid adic spaces

This is not correct in general. There are in fact two examples in Bosch's Lectures on Formal and Rigid Geometry p.61-63. Let me sketch the first one. While it uses rigid-analytic spaces, it can be ...
• 296