17
votes
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 "...
- 38k
16
votes
Accepted
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 ...
- 41.4k
16
votes
Accepted
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 ...
- 156k
12
votes
Accepted
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
11
votes
Accepted
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, ...
- 9,263
11
votes
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.
...
- 47.1k
8
votes
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 $\mathbb Q$.
...
8
votes
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. $\frac{4^m-1}{3} = 1 + 2^2 + 2^4 + \cdots + 2^{2m-2}$ ...
- 7,226
7
votes
Accepted
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.
...
- 47.4k
6
votes
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 ...
- 20.8k
6
votes
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-...
- 47.1k
6
votes
Accepted
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 ...
- 12.9k
6
votes
Accepted
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
5
votes
The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory
This issue already comes up in Tate's proof of the functional equation for zeta functions. The functional equation should come out of some version of Riemann-Roch and, for function fields, it does. (...
- 30.5k
5
votes
Birational Group Law
Thanks to nfdc23 for explaining the definition. There are counterexamples. For instance, begin with $S=\text{Spec}(R)$ for a discrete valuation ring $(R,\mathfrak{m})$. Let $\nu:\text{Spec}(\...
4
votes
Accepted
Understanding a valuation property of function fields
Let $m_0$ be the degree of $\mathfrak p$, so the splitting field of $\mathfrak p$ is the extension of $\mathbb F_q$ of degree $m_0$. The set of roots of $t^{p^m}-t$ contains this field extension if ...
- 22.5k
4
votes
Accepted
Henselian valued fields for characteristic $0$: a characterization
In general, i.e. for any valued field $(K,v)$, the implication $K = K_v \cap \overline{K}$ (or more precisely that $(K,v)$ have no immediate algebraic extension, i.e. that $(K,v)$ be algebraically ...
- 2,519
4
votes
Accepted
Rank 1 valuations that are not discrete on finite transcendental extensions of the rationals
Example: $val(X_1^{e_1}X_2^{e_2})=e_1 + e_2 \sqrt{2}$.
- 2,665
4
votes
In a CM field, must all conjugates of an algebraic integer lying outside the unit circle lie outside the same?
To expand on GNiklasch's answer, and analyse what you write as well: we always have (when complex conjugation is central in the Galois group) have
$\overline{\alpha^{\sigma}} = {\bar \alpha}^{\sigma}$ ...
- 44.4k
4
votes
Accepted
Recurrence for the sum
Let $n=m^tk$ where $m\nmid k$. Then $f(n)=m^t$.
Furthermore, if $t>0$, then $f(n/m)=m^{t-1}$ and $n-f(n/m)=m^{t-1}(mk-1)$. It follows that $a(n)=a(m^{t-1}k)+a(m^{t-1}(mk-1))$ and further by ...
- 34.3k
3
votes
Accepted
What is the definable functor associated to an algebraic scheme (model theory of valued fields)
This follows by elimination of imaginaries in algebraically closed fields. Given a finite affine cover of your scheme, one obtains a functor to Set by gluing the affine charts. This is indeed not ...
- 315
3
votes
Accepted
Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?
There is apparently not a very short answer to this. It is, I think, really not known whether this extension is elementary and it would be very surprising if it was known, since we do not even know if ...
- 311
3
votes
Accepted
Definition of model functions and their density in $C^0(X^\text{an})$
As alluded to in the question, the space $\mathcal{D}(X)_{\mathbb{Z}}$ is indeed the space of model functions arising from integral divisors.
Now, in order to deduce that $\mathcal{D}(X)_{\mathbb{Q}}$ ...
- 582
3
votes
Accepted
Valuation of congruent elements in a local division ring
$\newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}}$
Not necessarily.
Take $K = \Q_3 + \Q_3 i + \Q_3 j + \Q_3 ij$ with $i^2=-1$ and $j^2=3$, $r=2$, $x=y=j$, $a=3i$ and $b=3(1+i)$. Then $v(j)=1$, ...
- 5,382
3
votes
Accepted
'Smallest' subfield of the Surreals which is isomorphic to the Surreals as an ordered group
$\DeclareMathOperator{\Noo}{\mathbf{No}}$This might actually be a dead end.
This is because if $F$ is isomorphic as an ordered group to $\Noo$, then their value classes under natural ordered group ...
- 2,519
3
votes
Given a non-field local domain $R$, finding a dominating Valuation ring whose residue field is algebraic/finite extension of the residue field of $R$
Yes, an algebraic extension is always possible. If one applies Zorn's lemma not to
$$\{R\subseteq V\subseteq Q(R) \mid V\,\text{local}\}$$
but to
$$\{R\subseteq V\subseteq Q(R) \mid V\,\text{local} \...
- 9,650
3
votes
Recurrence for the sum
$$S(n)=\sum_{k=0}^{m^n-1} a(k)=\sum_{l=0}^{m-1}\sum_{k=0}^{m^{n-1}-1} a(km+l)$$....$(0)$
Using, the recurrence relations we simplify this to
$$S(n)=mS(n-1)+\sum_{k=0}^{m^{n-1}-1}a(km-f(k))$$.... $(1)$
...
- 1,755
3
votes
References on function fields over imperfect fields in positive characteristic
Not sure if this is what you have in mind, but the paper
Michael Szydlo, Elliptic fibers over non-perfect residue fields, J. Number Theory 104 (2004), no. 1, 75-99 (MR2021627)
is a detailed study of ...
- 47.4k
3
votes
Accepted
Higher-rank Archimedean valuations of $\mathbb{Q}$, does it exist?
Consider the following valuation:
$|\cdot|:\mathbb{Q}\to[0,\infty)^\mathbb{N}$ defined as
$|q|=(|q|_{p_i})_i$ where $(p_i)_i$
is the increasing sequence of all the prime numbers and $|\cdot|_{p_i}$ is ...
- 596
3
votes
For an element in the integral closure of an ideal $I$ - which power is in $I$?
Not a complete answer to your question, but the number $k_0$ (or $k_0-1$) is called the Briançon-Skoda number of $R$. For analytic planar curves, there is a formula for this number in
J. Sznajdman: ...
- 1,457
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