The valuation-theory tag has no wiki summary.

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $Y = A^n$. ...

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### 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 ...

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### 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) = ...

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### 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 ...

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### 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 $\Gamma$, ...

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### 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 ...

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### 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. ...

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### 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.
...

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### 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= \{ ...

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### 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 ...

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### 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 ...

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### 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 + ...

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### 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 ...

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### Henselization of valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks

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### 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 ...

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### 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 ?
...

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### 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 ...

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### 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?

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### 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 $v:K^{\times}\to\Gamma$ ...

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### 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?

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### 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 ...

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### 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($x-c=y$).

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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'. ...

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### Valuations and separable extensions

Let $R$ be a valuation ring containing a field $k$, with residue field $F$ and quotient field $K$. Assume $F/k$ is separable. Is $K/k$ separable?
I have convinced myself that (for a positive answer) ...

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### Why are extensions so heavily emphasized in valuation theory?

Whenever I read anything about valuations or things related to them (such as local fields) extensions always occupy a prominent position and a huge amount of effort is expended to derive results about ...

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### How exotic can DVRs be in the ring of rational functions over a local field?

Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$.
Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that ...

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### Complete extensions of valuations from Q to R.

This is somewhat related to the question and the answers here:
Is completeness of a field an algebraic property?
My question is (to which I believe the answer must have been known), does every ...

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### Are valuation rings regular?

This question is short, and to the point:
Valuation rings are certainly integrally closed, but are they regular?
The motivation is that I'm trying to understand the resolution of singularities of ...

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### Zariski-style valuation theory

I've been trying to read some of Zariski's older works, and I'm having some trouble getting into his mindset. I'd appreciate some help with this.
To quote Zariski (in "normal varieties and birational ...

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### Is a valuation domain PID when its maximal ideal is principal?

It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?

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### Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

A scheme is defined to be a sheaf which is locally isomorphic to the spectrum of a ring. The idea behind this is that given an affine coordinate ring of a variety over an algebraically closed field, ...

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### Improvements of the Baire Category Theorem under (not CH)?

The Baire category theorem implies that a nonempty complete metric space without isolated points must be uncountable. In many situations I have encountered, the "natural examples" of complete metric ...

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### Trivial valuation

I'm pretty sure trivial valuation over a field cannot be extended to a non-trivial one in a bigger field. Is there a simple way to show this without using the sledge hammer theorem on valuation ...

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### Existence of maximal totally ramified extensions of an arbitrary CDVF

Let $K$ be a complete, discretely valued field with (let's say) perfect residue field $k$. We have a unique maximal unramified extension $K^{unr}$ of $K$ and a unique maximal tamely ramified ...

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### Fiddling with p-adics

A paper I'm reading implicitly assumes the statement: Let $K_0$ be the completion of $\mathbb {Q}_ p^{un}$. Then any finite extension of $K_0$ is complete with residue field $\bar {\mathbb {F}} _p$. ...

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### Newton and Newton polygon

What did Newton himself do, so that the "Newton polygon" method is named after him?

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### An unfamiliar (to me) form of Hensel's Lemma

In his very nice article
Peter Roquette,
History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355,
Fields Inst. Commun., ...

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### Can the algebraic closure of a complete field be complete and of infinite degree?

Yes, this is yet another "foundational" question in valuation theory.
Here's the background: it is a well known classical fact that the dimension (in the purely algebraic sense) of a real Banach ...

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### What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

In modern valuation theory, one studies not just absolute values on a field, but also Krull valuations. The motivation is easy enough:
If $k$ is a field, a valuation ring of $k$ is a subring $R$ ...