For questions related to valuation functions on a field.

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106 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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86 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 ...
6
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120 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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94 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 ...
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46 views

extension of valuation rings

Assume $L$ is a finite Galois extension of $K$, and $R$ is a valuation ring on $K$ with maximal ideal $\mathfrak{m}$. Is it true that there are only finitely many valuation rings $(O,\mathcal{M})$ ...
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47 views

Motivation behind the Archimedean norm on number fields .

It is easy to justify the use of non-archimedean norms on number fields from an "inside view" as arising from the prime ideals and are therefore clearly useful a priori. However, it seems to me that ...
4
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171 views

Intuition behind “Non-Archimedean” — two senses of “non-archimedean”.

There appear to be two senses of the qualifier "Archimedean" for fields. One is for ordered fields, and one is for "valued fields" (fields with an absolute value function defined). In the first case, ...
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123 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 ...
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67 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}, ...
3
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36 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: $$ ...
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71 views

Valuations on $\Bbb Q(t)$

Ex. $2.3.3$ in Algebraic Number Theory by Neukirch is the following: Let $k$ be a field and $K = k(t)$ the function field in one variable. Show that the valuations $v_{\mathfrak p} $ associated to ...
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135 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: ...
3
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50 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 ...
3
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76 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 ...
3
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113 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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110 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 $0$ ...
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30 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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36 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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53 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) = ...
2
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16 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, ...
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50 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 ...
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34 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 ...
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89 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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43 views

The number of abolute value on $\mathbb{Q}(\sqrt{2})$

Let $|.|$ be the usual absolute value on $\mathbb{Q}$. The number of absolute value on $Q(\sqrt{2})$ extending |.| is 2 since $x^2-2=(x-\sqrt{2})(x+\sqrt{2})$ in $\mathbb{R}[x]$. Let ...
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0answers
91 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 ...
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82 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 ...
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0answers
56 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
57 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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17 views

Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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30 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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25 views

Compare Valuations

Let $K$ be a field complete with respect to a discrete valuation $v$. Let $K_s$ its separable closure, $w$ the unique valuation extending $v$, $(\mathcal{O}_{K_s}$, $\mathfrak{M})$ respectively the ...
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53 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 ...
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0answers
24 views

Characterization of valuation domains by means of their maximal ideal

I know the following theorem. Let $V, W$ be two valuation domains with the same fraction field $K$. Suppose $V$ and $W$ share the same maximal ideal. Then $V=W$. In other words, fixed a field ...
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27 views

Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
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0answers
43 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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27 views

Surjectivity of the derivation map on Washnitzer Algebra

Let $K$ be a non-archimedean field of characteristic zero and $||.||:K\to \mathbb{R}_{\geq 0}$ be its absolute value. Define the Washnitzer Algebra as: $$W_n=\{\sum_{u\in \mathbb{Z}_{\geq 0}^n} \in ...
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0answers
61 views

Extension of valuation

This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the ...
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0answers
37 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
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57 views

An example of a henselian valuation of rank 2.

I need to know simple examples of valuations of rank bigger than one. Please help me to concrete some examples of valuations of rank bigger than one with their valuation rings (specially henselian ...
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23 views

A valued field is complete iff its ring of intergers is complete

Let $K$ be a field, and let $v$ be a valuation defined on K, and let $O$ be the ring of integers, and $M$ be the maximal ideal. How to show that $K$ is complete (i.e $K$ is equal to its completion) ...
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0answers
174 views

Extensions of valuations

I'm trying to understand how a valuation $v$ of a field $K$ extends to an algebraic extension $L$. In chapter 8 of his book ANT, Neukirch first chooses a $K$-embedding $\tau : L \longrightarrow ...
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85 views

Henselization and 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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11 views

Are these value groups equal?

Assume that a field extension $L/K$ is finite, and K is a Henselian field with a exponential valuation $v$, and $w$ is an extension of $v$ to $L$. (If it is necessary, we can also assume that $L/K$ ...
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14 views

Simple Question about Valuations and Krull Rings

I have what is a very simple question about essential valuations for Krull rings. Before getting to the question, I'll give a sketch of the situation. Any help would be much appreciated. Suppose that ...
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0answers
19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
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23 views

Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?

Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...
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0answers
30 views

Units in discrete valuation ring

Let $k$ be field and $K=k(x)$ the field of rational function functions over $k$. Suppose that $p(x)\in k[x]$ is an irreducible polynomial. Define a map $v:K\to\mathbb Z$ by $v(p^r\frac{m}{n})=r$ where ...
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59 views

how can I prove this statement in local field theory?

Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer [edit] such that for each $x\in\mathbb{K}, ...
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81 views

Approach to build multivariate ranking system?

I have data of various sellers on ecommerce platform. I am trying to compute seller ranking score based on various features, such as 1] Order fulfillment rates [numeric] 2] Order cancel rate ...
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53 views

Smooth morphism and completion of DVR

Let $R$ be a Henselian discrete valuation ring with algebraically closed residue field and $\hat{R}$ its $m$-completion, where $m$ is the maximal ideal. Is it true that the induced morphism ...