For questions related to valuation functions on a field.

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Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
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1answer
28 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
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1answer
23 views

Given two extensions of $\left| \cdot \right|_p$ to $\Bbb{C}$, do the subsets of elements of absolute value $1$ coincide?

It is known that $\Bbb{C}$ is isomorphic as a field to $\Bbb{C}_p$, the completion of $\bar{\Bbb{Q}}_p$ with respect to $\left|\cdot\right|_p$. Clearly, given two such isomorphisms $\varphi_1$ and ...
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112 views

Determine all discrete valuations on $\mathbb{C}(x)$.

To clarify, for a field $K$, a valuation $v$ on $K$ is a map $v:K^{\times}\to G$ for $G$ an ordered group (written additively) such that for any $a,b\in K^{\times}$: 1) $v(ab)=v(a)+v(b)$; 2) ...
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1answer
32 views

A tool to prove Baby Ax-Kochen principle

I have been reading a lecture notes on model theory of valued fields written by Lou van den Dries. This is the question I have: It is known in this lecture that: Let $R$ be a local ring, with $t\in ...
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1answer
20 views

Cauchy sequence in a valuation ring

From Janusz's book algebraic number fields, chapter 2. Let $K$ be a complete field with respect to a non-Archimedean absolute value $|\cdot|$. Let $R$ be its valuation ring with maximal ideal ...
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1answer
17 views

Equivalence of norms in finite dimension over complete fields is true, but false for finite rank modules over complete rings

We know that if $k$ is complete valued field and $V$ a finite dimensional vector space then all norms on $V$ are equivalent. (The field is not necessarily of characteristic $0$ and its absolute value ...
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1answer
26 views

Extension of a discrete valuation on a complete field

Let $K$ be a complete field w.r.t discrete absolute value $|\cdot|_K$, $\mathcal O_K=\{x\in K:|x|_K\leq 1\}$. $L$ is an extension field with $[L:K]< \infty$ and let $\mathcal O_L$ be the ...
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55 views

Roots of unit in a DVR

Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$. I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the ...
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1answer
37 views

Valuation rings are conjugate

Let $F/K$ be a finite Galois extension where $(K,v)$ is a valued field (i.e. $v$ is a valuation on $K$). Let $w_1,w_2$ be extensions of $v$ to $F$. Then, we have associated valuation rings $O_{w_1}$ ...
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26 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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1answer
36 views

Number of places over a number field

Let $\mathbb{K}$ a finite extension of $\mathbb{Q}$ and $\mathcal{O}_\mathbb{K}$ its ring of integers. Assume $\mathcal{O}_\mathbb{K}=\mathbb{Z}[\alpha]$, that is generated as a ring by a single ...
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1answer
17 views

Definitions of valuations in terms of totally ordered group

Wikipedia gives a definition of valuations involving abelian totally ordered groups. So far I have only seen valuations taking values in the real numbers. Is there a reason for this generalization?
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17 views

Reference request for valuations in algebraic number theory

I am going through the book "Primes of the form $x^2$ + n$y^2$" and I understand the background material about algebraic number fields and class numbers(from Ireland and Rosen). However, I do not have ...
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1answer
32 views

Negative degree valuation: valuation ring and its maximal ideal

I know that the $v: f \mapsto -\deg(f)$ is a discrete valuation on the field of complex rational functions $\mathbb{C}(X)$ (the quotient field of $\mathbb{C}[X]$). The valuation ring $\mathcal{O}_v$ ...
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43 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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40 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 ...
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0answers
13 views

Determine the valuation of $\rho$ with $F = k(x,\rho)$ at the only place above $(\infty)$.

Let us assume that we have the following setup. Let $F=k(x,\rho)$ be an algebraic function field with $$f(x,y) = y^n+a_1y^{n-1}+\cdots+ a_iy^{n-i}+\cdots+a_n \in k[x][y]$$ irreducible in $y$, and ...
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0answers
26 views

Computing other valuations of a field

Assume $k$ is an algebraically closed field, and $x$ and $y$ are indeterminant over $k$, I know valuations of $F$, the field of fractions of the ring $A=\dfrac{k[x,y]}{I}$ where $I$ is an ideal of ...
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17 views

How to construct a unique valuation for $k\left(T_{i}\right)_{i\in\mathbb{N}}$ in $\mathbb{Z}^{\left(\mathbb{N}\right)}$?

Let $k$ be a field and $\left(T_{n}\right)_{n\in\mathbb{N}}$ indeterminates over $k$. Let $K=k\left(T_{n}\right)_{n\in\mathbb{N}}$ and $\varGamma:=\mathbb{Z}^{\left(\mathbb{N}\right)}$ the abelian ...
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31 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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1answer
41 views

Completion of a number field at a complex embedding

Sorry if this question has been asked before. Let $K$ be a number field of degree $n>1$ and $\sigma:K\hookrightarrow \mathbb C$ a complex (non real) embedding of $K$ in $\mathbb C$ giving the ...
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1answer
32 views

Is it true that an equivalent 'absolute value' is an absolute value?

I've a very basic question on absolute values on fields. If $K$ is a valued field with absolute value $|- |:K\to \mathbb R_{\geq0}$ then is the map $|-|':K\to \mathbb R_{\geq0}$ defined by ...
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1answer
42 views

Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
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36 views

Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
3
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1answer
34 views

valuation ring, completeness

Perhaps a trivial question: is there an example of a field $K$ and a valuation $v$ on $K$ such that the following holds: $K$ is not complete (with respect to the valuation topology) The valuation ...
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1answer
22 views

Dimension of quotients of a discrete valuation domain

I'm learning some properties of discrete valuation rings (DVR's further for geometrical use). By the way, a domain $R$ is said to be a DVR if there exists the so called uniformizing parameter $t$ such ...
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2answers
62 views

It's true that a valuation ring $R$ in the quotient field of a normal ring $A$ contain $A$?

Let $A$ be a finitely generated $k$-algebra ($k$ algebraically closed) of dimension one, integrally closed in its quotient field $K$. Let $R\subseteq K$ be a valuation ring. It's true that $A\subseteq ...
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1answer
46 views

Can a $p$-adic field admit a different valuation?

Let $L/\mathbb{Q}_p$ be a finite extension. Question: Is it possible for $L$ to admit a henselian valuation with residue characteristic $q \not=p$? I would think surely not, but I can't see a ...
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1answer
112 views

The role of valuation rings in algebraic geometry

I am familiar with basic algebraic geometry in the tradition of Hartshorne's book. Discrete valuation rings appear there in the criteria for separatedness/properness, and are used to define the order ...
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1answer
93 views

Infinite primes of a number field

Let $K$ be a number field. I know that to each real and to each complex conjugate pair of embeddings of $K$ there corresponds exactly one prime (equivalence class of absolute values) of $K$. How do I ...
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2answers
112 views

Is every local ring a valuation ring?

Is every local ring a valuation ring? I know the answer is no and the first example comes to my mind was as following (I started with smallest fields, as $\mathbb{Z}_2$ and $\mathbb{Z}_3$ are ...
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0answers
45 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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0answers
36 views

Question about valuation rings of a rational function field

Let $k$ be a field and $f \in k[x,y] \setminus k$. Write $E=k(f)$ and $L=k(x,y)$. Assume that there exists a $g \in k(x,y)$ such that $L = E(g)$. Suppose there exists a valution ring $\mathcal{O}$ ...
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65 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 ...
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1answer
40 views

Noetherianity of valuation ring and valuation being discrete

I need a hint for left to right part of the following: Let $K$ be a valued field with $\nu$ and $\mathcal{O}_\nu$ be its valuation ring. Then, $\mathcal{O}_\nu$ is Noetherian if and only ...
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2answers
47 views

Discrete valuations of the rational numbers

I'm trying to find every discrete valuation on the field of rational numbers. If $a\in \mathbb Q$, we can write $a=p^j\frac{x}{y}$, where $p$ is a prime number and $p\nmid x$ and $p\nmid y$. We can ...
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1answer
43 views

Places of this extension

I'm reading this book. I'm trying to find the degree of the places of the extension $\mathbb C(X)\mid\mathbb R$. I know the places of the extension $\mathbb R(X)\mid\mathbb R$ and I've already ...
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1answer
65 views

real closure of an archimedean field

my question is: Is an archimedean field dense in its real closure? I know that in the non-archimedean case, this does not have to be true (e.g., rational fucntions). Thanks!
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1answer
61 views

When Is there no Local Power Integral Basis?

Let $A$ be a Dedekind domain, $K, L, B$ the usual designations of $A$'s quotient field, a finite separable extension of $K$, and integral closure of $A$ in $L$ respectively. If $\alpha \in B$ ...
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1answer
40 views

Extensions of nonarchimedean valuations

This is a question from Janusz 'Algebraic Number Theory'. Let $R$ be a DVR with maximal ideal $\mathfrak p=\pi R$. Let $K$ be the quotient field of $R$ and $\mid\cdot\mid_{\mathfrak p}$ the ...
3
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1answer
51 views

Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
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38 views

Hensels Lemma in many variables

Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$. Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable ...
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1answer
38 views

tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :) I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting: $k$ an ...
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0answers
33 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
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1answer
104 views

Question on extensions of discrete valuation fields

Let $F$ be a discrete valuation field. Let $L$ be a finite extension of $F$. Let $L=F(\alpha)$ where $\alpha$ belongs to ring of integers of $L$, denoted by $O_L$. Is it always true that ...
2
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0answers
31 views

automorphism of $\Bbb{Q_p}$closed [duplicate]

How to show that there does not exist an automorphism of $\Bbb{Q_p}$ except identity. Please help. Is any automorphism of $\Bbb{Q_p}$ already continuous.?
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1answer
92 views

Question on complete discrete valuation field.

Let $F$ be a complete discrete valuation field and $f(X) = X^n + a_{n-1}X^{n-1} +\cdots+ a_0$ is an irreducible polynomial over $F$. How to show that a) $ v(a_0) > 0$ implies $v(a_i) > 0$ for ...
2
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0answers
41 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 ...
4
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1answer
52 views

Why is the order of a prime element well-defined?

This is in relation to the $p$-adic valuation on the field of fractions $F$ of an integral domain $D$. The idea is that for each $x \in F$ there is a unique maximal $k$ such that $x = p^k u v^{-1}$ ...