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Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

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280 views

An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
7
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0answers
299 views

Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
6
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0answers
159 views

(Sanity check) the adic spectrum $\operatorname{Spa}(\mathbb{Z}, \mathbb{Z})$

I am following Scholze's and Weinstein's notes on $p$-adic geometry on http://www.math.uni-bonn.de/people/scholze/Berkeley.pdf, example 2.3.6 (page 18 as of this moment bar any updates). $\...
6
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84 views

If $\xi$ is unramified at the valuation $v$, does the associated homogeneous space $C/K$ have a $K_v$-rational point?

I'm looking at the following Corollary/Remark $(4.4-4.5)$ in Chapter X of Silverman's Arithmetic of Elliptic Curves: Let $\phi:E/K\to E'/K$ be an isogeny defined over $K$, and $S$ a finite set of ...
6
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152 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
6
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0answers
108 views

Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
5
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142 views

Is the topology of the p-adic valuation to the unramfied extension discrete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(...
5
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69 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
4
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0answers
67 views

In valued fields (of rank > 1), is the p-adic completion of the algebraic closure still algebraically closed?

Let $K$ be a valued field and assume its valuation ring $\mathcal{O}_K$ is $p$-adically separated. If we assume further that $K$ is algebraically closed and we take the $p$-adic completion, is the ...
4
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65 views

How to write the Adeles over $\mathbb{Q}(i)$?

What do these adeles look like over $\mathbb{Q}(i)$? These are simply fractions where the numerator and denominator are allowed to have the number $i = \sqrt{-1}$. The elements look like: $$ \left\...
4
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584 views

Archimedean places of a number field

Let $K$ be a number field with an Archimedean absolute value $|\cdot |$ and let $\bar{K}$ be the completion of $K$ wrt this valuation. Then $\bar{K}\cong \mathbb R $ or $\mathbb C$. My question is: ...
4
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142 views

Transfinite induction and valuation rank

In Engler's valued fields, exercise 3.5.2 goes as follows Construct valuations on $\Bbb{C}$ of rank $\kappa$ for every cardinal $\kappa \leq 2^{\aleph_0}$. The idea behind this (for any countably ...
3
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35 views

Why does the image of $ord_p$ form an additive subgroup of $(1/n)\mathbb Z$?

Let $K$ be a field extension of the p-adic rationals $\mathbb Q_p$. The image of $K^\times$ under the valuation map $ord_p(x)=-\frac 1 n\log_p|\mathbb N_{K/\mathbb Q_p}(x)|_p$ is contained in $(1/n)\...
3
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97 views

Fraction field of $A[[t]]$

Let $A$ be a complete discrete valuation ring. (For example $A$ is the completion of the local ring of a curve at a point). What is the fraction field of $A[[t]]$? Is it $\operatorname{Frac}(A)((t))$?...
3
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55 views

On $A$ algebra homomorphisms $A[[X_1,…,X_n]]\to Q(A)$, where $A$ is a complete DVR

Let $(A,\mathfrak m)$ be a complete Discrete Valuation Ring (complete w.r.t. the $\mathfrak m $-adic topology) with fraction field $K$. Let $\phi : A[[X_1,...,X_n]]\to K$ be an $A$-algebra ...
3
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79 views

Extension of complete discrete valuation fields and coefficient field

Let $L$ and $K$ be two complete discrete valuation fields of equal characteristic $0$. Assume that an embedding $K\subset L$ is fixed and let $F$ be the algebraic closure of $K$ inside $L$. Can you ...
3
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26 views

Residue division ring of a division ring over complete discrete valuation field

Let $K$ be a complete discrete valuation field with valuation ring $O_K$ residue field $k$. For a finite dimensional division ring $D$ over $K$ with center $K$, we can extend the valuation of $K$ to $...
3
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0answers
39 views

Absolute value of a generator of the different

Let $K$ be an algebraic function field over $\mathbb F_q$, in other words $K$ is a finite extension of $\mathbb F_q(t)=F$; furthermore fix a valuation $v$ on $K$. Consider the relative different ...
3
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214 views

Discrete valuation on $\mathbb{Q}(X,Y)$ such that the residue field is $\mathbb{Q}$

I tried multiplicity of zeros and poles because this works for $\mathbb{Q}(X)$. I guessed that this would be the same for multivariable cases, but it looks more complicated and I don't know how to ...
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149 views

Centers 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 $D_{\tau}...
3
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0answers
64 views

Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that $w$...
3
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104 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field $k(x,y)...
3
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146 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
3
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105 views

How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
3
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0answers
175 views

Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
3
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0answers
83 views

A trace formula for an extension of complete valued fields

I'm trying to understand the following fact. Proposition. Let $L/K$ be a finite separable extension of complete valued fields, with respective valuations $w$ and $v$, valuation rings $\mathcal{O}_L$ ...
2
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0answers
48 views

Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
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42 views

Are Valuations on algebraic extensions of an henselian field unique?

The definition of henselian is: A valued field $(\mathbb{K},\nu)$ is said to be Henselian if for any algebraic extension $\mathbb{L}$ of $\mathbb{K}$ there is a unique valuation $\tilde{\nu}$ on $\...
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31 views

The valuation attached to a smooth point of an algebraic variety

I am interested in proving the following result: Let $k$ be an algebraically closed field, $X$ a normal integral variety over $k$ and $x\in X$ a closed point. Write $\mathfrak{m}_x$ for the ...
2
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0answers
94 views

On the maximal unramified extension of $\mathbb{Q}_{p}$ of a given degree.

I'm stucked with a theorem withouth proof saw in a Book about G-Functions by Dwork, I will appreciate any hint, also I provide a ''proof'' of that theorem, but a feel that is too ''bla bla'' and I ...
2
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0answers
78 views

henselian valuation field is algebraically closed in its completion?

Let K be a henselian rank 1 valuation field, is it true that K is algebraically closed in its completion? This question is a special case of Relation of Fontaine-Wintenberger theorem and tilting ...
2
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48 views

Criterion of separatedness and propernesss

This is Exercise 4.5 of Hartshorne Chapter II Let $X$ be an integral scheme of finite type over a field $k$, having function field $K$. We say a valuation of $K/k$ has center $x$ on $X$ if its ...
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113 views

How a finite extension of transcendence degree $1$ extensions induces a morphism of curves

Let $k$ be an algebraically closed field. Throughout, by curve I mean integral, nonsingular, dimension $1$ scheme proper over $k$. In particular I'm assuming we're dealing with complete curves in the ...
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30 views

Growth rates of matrices

Let $D$ be a discrete valuation ring, possibly non-commutative, with uniformiser $\pi$, and let $Q=M_n(Q(D))$, where $Q(D)$ denotes the ring of quotients of $D$. Let $v$ be the extension of the $\pi$-...
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37 views

The proof of $A/m^n$ isomorphic to $\hat {A} /\hat {m}^n$

$A$ is a discrete valuation ring, $m$ its maximal ideal, $\hat A, \hat m$ their completion respectively. In J.S. Milne's Algebraic Number Theory, p. 116: Why the openness implies the surjective ...
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190 views

valuation vs exponential valuation

I'm having some trouble to understand the concept of valuations (and exponential valuations) and would be very grateful if one of you could tell me whether the following is right: First of all, there'...
2
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0answers
92 views

An application of valuative criterion for properness

I want to show: Let $R$ be a Dedekind domain with fraction field $K$. Let $X$ be a scheme and $X \to \operatorname{Spec}R$ a proper morphism. Show that the natural map $$X(\operatorname{Spec}R)\to ...
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59 views

Does every coset in a ring of integers contain a totally positive element?

Let $K$ be a number field with ring of integers $\mathcal O_K$, let $\mathfrak m$ be an ideal in $\mathcal O_K$ and let $a \in \mathcal O_K$ such that $(a, \mathfrak m) = 1$. Does there necessarily ...
2
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0answers
88 views

Principal local Artinian ring is a quotient of discrete valuation ring.

I have seen here the following statement: Let $R$ be a principal local Artinian ring. Clearly the quotient of a discrete valuation ring is such a ring; conversely it is not difficult to show that ...
2
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0answers
99 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 ...
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0answers
116 views

Valuations above $\mathbb{Q}(\alpha)$

Let $K$ be a field, we say that a function $\nu:K\to \mathbb{Z}_{\geq 0} \cup \lbrace \infty \rbrace$ is a valuation above $K$ if then followings hold for each choose of $x,y \in K$: 1 $\nu(x) = \...
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0answers
75 views

Rational points of an open subset of a group scheme

Let $K$ be an algebraically closed valued field with valuation ring $\mathcal{O}$ (assume the valuation is non-trivial). Let $G$ be a group scheme over $\mathcal{O}$, $g\in G(\mathcal{O})$ and $V\...
2
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0answers
53 views

question on proof in Frohlich/Taylor regarding extension of a discrete absolute value on a complete field

I am having some trouble understanding a proof in Fröhlich and Taylor, pages 105-106. There $K$ is a complete field with a discrete absolute value $u$ $\mathfrak o,\mathfrak p$ is the valuation ring, ...
2
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0answers
72 views

Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
2
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0answers
57 views

Geometric structure on the set of valuation rings of a field

Let $K$ be a field. Let $\mathcal{O}_K$ be the intersection of all valuation rings with quotient field $K$. Can someone give an example of a field $K$ in which we don't have a bijection of sets: $$ \...
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0answers
99 views

Computing a valuation of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are transcendental over $k$. I want to compute the valuation ring of $F$, the field of fractions of the ring $A=k[x,y]/I$, where $I=\langle ...
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0answers
123 views

Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
2
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0answers
69 views

A question on a sum of valuations

Let $A$ be a discrete valuation ring of characteristic zero. Let $v$ be the valuation on $A$. Let $I$ be a finite index set and $d_i$ a positive integer for all $i$ in $I$ and define $$ d:= \sum_{i \...
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vote
0answers
19 views

Normal basis of finite extension of a complete DVR

Let $R$ be a complete discrete valuation ring, and $S$ be a finite extension such that the associated residual field extension is separable. Then, why is it possible to choose a normal basis in powers?...
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vote
0answers
45 views

Valuation of a number field element over a prime ideal in an order

Having read https://mathoverflow.net/questions/144671/number-field-sieve-for-factorization-with-non-monic-non-linear-polynomial-cant I stumbled on a problem I can't prove. Most of the questions posed ...