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6
votes
0answers
115 views

Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$. We usually call it $\mathbb{C}$, but by this we impose a ...
2
votes
0answers
49 views

Relation between ramification index and length of filtration of ramification groups

Given a complete valued field $K$ with a discrete value group $\mathbb{Z}$, consider a totally ramified finite Galois extension $L$ of $K$ with its Galois group $G$. Let $O_L$ be the valuation integer ...
0
votes
0answers
67 views

Extension of a valuation on a function field

Let $K$ be a field, and $K(x)$ the field of rational functions over $K$. Consider the degree valuation $v$ on $K(x)$, That is $v\left(\frac{f(x)}{g(x)}\right)=\deg(g)-\deg(f)$. So for every $f(x)\in ...
6
votes
2answers
211 views

Examples of NIP fields of characteristic $p$

Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the ...
4
votes
1answer
264 views

Completion of a finite field extension is also finite?

Let $(L,w)/(K,v)$ be a finite extension of valuation fields, and let $L_w$, $K_v$ be the respective completions of $(L,w)$, $(K,v)$. Is the field extension $L_w/K_v$ finite? For nonarchimedean ...
1
vote
1answer
86 views

Separable extensions of henselian fields

Let $(k,v)$ be a henselian field, with $\mathcal{O}$ and $\bar{k}$ being respectively its valuation ring and its residue field. If $K/k$ a finite separable field extension (on which $v$ thus extends ...
0
votes
1answer
173 views

Model over DVR for smooth projective curves

Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...
1
vote
2answers
273 views

Subgroup of Projective general linear group on complete discrete valuation ring

Let $R$ be a complete dvr and $k$ its residue field of positive characteristic. Let $H$ be a finite subgroup of $PGL_2(k)$ such that the order of $H$ is prime with $char(k)$. Is there some ...
0
votes
0answers
78 views

center of divisorial valuations on toric varieties

Suppose we are given a divisorial valuation $\mathcal{v}$ on a smooth toric variety $X_{\Sigma}$ (for a fan $\Sigma$), i.e. $\mathcal{v}$ is the valuation induced by a torus invariant divisor ...
1
vote
1answer
186 views

Completion of discrete valuation ring

Let $R$ be an excellent, Henselian, discrete valuation ring with algebraically closed residue field and $\hat{R}$ be the completion of $R$. If I understand correctly, the residue field of $\hat{R}$ is ...
0
votes
0answers
169 views

Extension of a complete discrete valuation ring

My question came when I was reading the famous Tate's paper on $p$-divisible groups. At the beginning of chapter $(2.4)$ he cites this fact as obvious. If you take a complete discrete valuation ring ...
3
votes
1answer
168 views

On conductors, levels and traces on quaternion algebras

I am currently working on level issues in the division central simple algebra case, say $D$ over a local non-archimedean field $F$ (e.g. $\mathbf{Q}_p$). Let say that $\mathcal{O}_D$ and ...
8
votes
0answers
262 views

Is it decidable whether a finite type scheme is proper?

Let $k$ be a field and let $X$ be a finite type scheme over $k$, explicitly given by finitely many affine patches which are $\mathrm{Spec}$ of finitely generated $k$-algebras, glued along other affine ...
9
votes
2answers
462 views

How did height in algeb. number theory/elliptic curves started?

Maybe this is obvious but it isn't to me yet. What is the history of heights used in say points of the project plane over a number field or of elliptic curve over a number field? I would guess people ...
3
votes
0answers
143 views

Extension of the product formula for valuations to a simultaneous completion

It is well known that $\mathbb{C}$ and $\mathbb{C}_p$ are "algebraically" isomorphic (that is, ignoring the topology), but an isomorphism depends on the axiom of choice and there is no canonical way ...
6
votes
3answers
408 views

Finite extension of local fields

Can a (higher) local field have uncountably many finite (seperable) extensions?
0
votes
2answers
257 views

Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
2
votes
0answers
126 views

seek another proof of a result in Fourier analysis

It was proved on page 26 of this note the following result: Let $\xi$ be an algebraic number that is not a root of unity, then there exists an $n_0\geq 0$ with the property that ...
1
vote
1answer
257 views

Maximal unramified extension and inertia group for separable closure

I have a problem in understanding the inertia group of an infinite extension. I am studying it in this context. Let $K$ be a field, $v$ a discrete valuation on $K$, and $\mathcal{O}_v$ the discrete ...
2
votes
0answers
125 views

Jacobian Conjecture, Cubic-Keller maps

I have recently read an interesting article about the Jacobian Conjecture, in particular the reduction to the case $f(x) = x + A(x)^3$. I was wondering about codimension one divisors on $Y = A^n$. ...
1
vote
0answers
48 views

Saturation of a subalgebra over the Tate-algebra inside the power series ring

Let $A$ be a discrete valuation ring and $\pi$ a uniformizer. Over $A$ we consider the Tate-algebra $$A\langle t \rangle =\{ f=\sum_{n=0}^\infty a_nt^n \mid a_n\in A, \lim_{n\to \infty} \lvert ...
2
votes
0answers
143 views

Valuations given by flags on a variety and valuations of maximal rational rank

I am interested in valuations on a function field $K=k(X)$ of some say smooth, projective $k$-variety $X$ of dimension $n$, where $k$ is some (algebraically closed) field (that implies trdeg$(K/k) = ...
9
votes
1answer
350 views

Analogy between Lagrange's Theorem and Rank-Nullity Theorem?

One can view view Lagrange's Theorem $$|G/H|=|G|/|H|$$ and the Rank-Nullity Theorem $$\dim(V/U)=\dim(V)-\dim(U)$$ as directly analogous. Does anyone know a high-level explanation of this analogy? I ...
2
votes
0answers
167 views

Reference for “approximately henselian” valued fields

I need some valuation theory in a paper I’m working on. This is not quite within my area of expertise, and I’d like to make the terminology right. A valued field $(K,v)$ with value group $\Gamma$, ...
4
votes
0answers
210 views

Discrete valuations for which Abhyankar inequality is strict

The background to my question, in a nutshell, is: If $k$ is a field and $X$ a $k$-variety, i.e. an integral, separated, finite type $k$-scheme, which discrete rank $1$ valuations on $k(X)$ come from ...
0
votes
1answer
105 views

Complete D.V.R's That have different characteristic than the residue field

I'm working through Local Fields by Serre and am stumped by something that he thinks should be obvious. Let $A$ Be a complete D.V.R with uniformizer $\pi$ and $\overline{K}$ be it's residue field. ...
4
votes
3answers
219 views

Reference for Ostrowski's 1916 Theorem?

I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations. ...
2
votes
0answers
204 views

Asymptotics vs Puiseux series

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= \{x_i\} \lt Y= \{ ...
1
vote
0answers
136 views

Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...
0
votes
1answer
154 views

Finite extension of valuation

This is a similar question from the book "Valued Fields by Antonio J. Engler and Alexander Prestel, Springer, 2005 " page 82, Exercise 3.5.4.(b). Let $(K_{1}, V_{1})\subseteq (K_{2}, V_{2})$ be ...
0
votes
0answers
211 views

Pseudo-cauchy sequence and valuation

Let $k$ be a field and $x$ is transcendental over $k$. Can we construct a pseudo-cauchy sequence $(a_{i})$ convergent to $x$ with each $a_{i}$ is algebraic over $k$ and $k(a_{i})\subseteq k(a_{i + ...
0
votes
1answer
121 views

Henselization of valued field

Let $(K, \nu)$ be a valued field and $x$ is transcendental over $K$. Is there exist a henselian extension of $(K, \nu)$ in between $(K, \nu)$ and $(K(x), \nu^{'})$ where $\nu^{'}$ is an extension ...
1
vote
1answer
257 views

Henselization of valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
1
vote
2answers
245 views

Algebraic maximal extension and algebraic closure

Let $K$ be a valued field. We say that $K$ is algebraic maximal if any algebraic extension of $K$ has either a bigger value group or a bigger residue field. Under which condition is an algebraic ...
3
votes
1answer
126 views

when is the property “being algebraically maximal” a first order property ?

A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field. Do you know for which field this is a first order property ? ...
8
votes
1answer
419 views

Valuations on tensor products

Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equivalent valuations on ...
1
vote
0answers
163 views

Extending commuting endomorphims of a complete discrete valued field to the algebraic closure?

Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?
6
votes
1answer
227 views

When is a valued field second-countable?

Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial). The valuation $v:K^{\times}\to\Gamma$ ...
1
vote
0answers
179 views

uniqueness of a limit of a pseudo convergent set

Is there an example of valued field in which any pseudo convergent set has a limit and such that this limit is unique?
2
votes
1answer
295 views

Quotient field extension for an incomplete DVR

Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q(R)$, the fraction ...
1
vote
2answers
389 views

Produce an irreducible polynomial that can't be proved irreducible by using Eisenstein [closed]

give An example of an irreducible polynomial that cannot prove it by using the Eisenstein criterion even with the use of all linear change variable($x-c=y$).
3
votes
1answer
765 views

The space of valuations of a function field

Hello, I'm looking for someone who can help me to understand Zariski's theory of valuations. First I outline the theory: we take a field $K$ which is a finitely generated transcendent extension of ...
5
votes
2answers
482 views

If the discriminant of a binary quadratic form has high valuation, is the form “almost a square”.

For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...
0
votes
1answer
363 views

Relating $p$-adic Valuations of Elements in $\mathbb{C}$ and $\mathbb{C}_p$

Let $K = \mathbb{Q}(\theta)$, where $\theta$ is a root of an irreducible polynomial $g \in \mathbb{Z}[t]$. Fix a rational prime $p$. Let $\theta^{(1)}, \ldots, \theta^{(n)}$ be the roots of $g$ in ...
4
votes
1answer
833 views

Does totally ramified extension really exist?

The answer is certainly "Yes", but this is the problem I met in Algebraic Number Theory by Neukirch. I guess that I must be doing something wrong, since otherwise I will get the statement "There are ...
4
votes
0answers
326 views

Sheaf Cohomology on Zariski-Riemann Spaces

Can sheaf cohomology on the Zariski-Riemann spaces give some sort of classification for field extensions (even just for function fields)? If not, are there any significant or useful results (e.g. for ...
4
votes
3answers
733 views

Chevalley's valuation extension theorem and the axiom of choice

Hello, Do we know if the axiom of choice is needed for Chevalley's valuation/place extension theorem (i.e. the theorem that states that for every valued field and a field extension, one can extend ...
4
votes
0answers
497 views

Tropical Properties From Algebraic Geometry

What properties of tropical geometry (Starting from a valued Field) can be proven to be true using their analogue in algebraic geometry? For example, using the valuation on the Puiseux series ...
1
vote
1answer
483 views

Completeness of Algebraically Closed Valued Fields(ACVF) Theory

One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,*, U, \mid )$. In ...
2
votes
0answers
524 views

Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. ...