For questions related to valuation functions on a field.

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37 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 ...
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1answer
15 views

Simple extensions of local fields

Let $L/K/\mathbb Q_p$ be finite extensions of local fields and let $v_L$ and $v_K$ be normalised discrete valuations on $L$ and $K$ respectively. My question is quite a general one: If the ...
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1answer
18 views

Maximal ideal of valuation

Let $\nu : K \rightarrow G \cup \{\infty\}$ be a map defined by, where $G$ is a totally ordered group and $g < \infty$ for all $g\in G.$ $\nu(a) = \infty$ if and only if $a = 0,$ $\nu(a + b) ...
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1answer
27 views

Two dimensional valuation domain

Let $R$ be a two-dimensional valuation domain with prime ideals $0 \subset P \subset M$ and value group $G=\Bbb Z \oplus \Bbb Q$. Then $M^2=M$ and $P^2 \neq P$. Why $M^2=M$ and $P^2\neq P$? Can we ...
2
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0answers
43 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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54 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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1answer
83 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
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0answers
20 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
23 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 ...
2
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1answer
27 views

Valuation of a particular element

I am tying to compute the valuation of a particular element of $\mathbb{Q}_p$. I am trying to compute $\operatorname{val}_p(P)$ where $P=\frac{\log(1+p^2)}{\log(1+p)}$ and $\log$ is the $p$-adic ...
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68 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 $R$ ...
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0answers
34 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 ...
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0answers
32 views

DVR and its fraction field

Let $k$ be a complete discrete valuation field with algebraically closed residue field. We know that its maximal unramified extension $k^{\mathrm{unr}}$ need not be complete. But can the ring of ...
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3answers
38 views

Extension of $|\cdot|_\infty$ on $\mathbb R$ to $\mathbb C$

Let $|\cdot|$ be the usual absolute value on $\mathbb C$. My question is: Is the only extension of $|\cdot|$ on $\mathbb R$ to $\mathbb C$ $|\cdot|$ itself? I'm not sure about the uniqueness. I ...
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1answer
60 views

If $\mathcal O_P(C)$ is a DVR, then $P$ is non-singular

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$. I would like to prove if $$\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$$ is a DVR, then $P$ is non-singular, i.e., the ...
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1answer
95 views

How to prove $\mathcal O_P(C)$ is a DVR for $P$ non-singular?

Let $C$ be an irreducible curve over $\mathbb A^2$ and $P\in C$ a non-singular point. I want to prove that $\mathcal O_P(C)=\{f\in k(C)\mid f=a/b, b(P)\neq 0\}$ is a DVR. I've already proved that ...
2
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1answer
20 views

Archimedean completion of a number field

Let $K$ be a number field. I want to show that Every Archimedean absolute value of $K$ is equivalent to the absolute value $|x|:=|\sigma(x)|_\infty$ for $x\in K$ where $\sigma$ is an embedding of ...
3
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0answers
50 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: ...
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0answers
45 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 ...
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0answers
40 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
23 views

Is $\mathcal O^{\ast}/U^{(n)}\cong (\mathcal O/\mathfrak p)^{\ast}$ as topological groups?

I have the following: $K$ is a field with discrete valuation $v$, $\mathcal O$ its valuation ring and $\mathfrak p$ the maximal ideal and $U^{(n)}=1+\mathfrak p^n$ the $n$-th unit group for $n\geq 1$. ...
3
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1answer
56 views

How to prove that any infinite algebraic extension of a complete field is never complete?

My first idea is using Baire category theorem since I thought an infinite algebraic extension should be of countable degree. However, this is wrong, according to this post. This approach may still ...
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0answers
25 views

Definable valuation ring

If $(K,v)$ is henselian and $O_v$ is $\phi$-$\text{definable}$, why do I have that if $L\equiv K$ (in the language of ring) then $L$ admits a non-trivial henselian valuation ring? I understand if ...
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2answers
40 views

A extending the p-adic valuation to a quadratic extension of $\mathbb{Q}_p$

I'm trying to solve the following problem. Prove that, if $d \in \mathbb{Z}_p$ is non-square, then $|a + b \sqrt{d}|p = |a^2 − b^2d|^{1/2}_p$ , for any $a, b \in \mathbb{Q}p$, defines a ...
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0answers
29 views

unramified extension of valued fields

I came across the following exercise: Let $M$ be a valued field with subfields $E$ and $L$, and suppose that $L$ is finite over some field $K\subseteq L\cap E$. Show that $EL/E$ is unramified if ...
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1answer
24 views

Why is the separable closure of a field in it's completion not complete.

I am currently reading "Algebraic Number Theory" by Neukirch and I am a bit confused by what is here. It is on page 143, it states this: $(K,v)$ is a nonarchimedian valued field and ...
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1answer
33 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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0answers
125 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
296 views

Two discrete valuation rings, one contained into another

Let $A$ and $B$ be discrete valuation rings of the same field of fractions. Suppose $A \subset B$. Then $A = B$? I came up with this problem when I was reading van der Waerden's Algebra. The ...
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1answer
36 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
25 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
20 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
32 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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0answers
58 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
39 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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0answers
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
39 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
20 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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1answer
43 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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49 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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42 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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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
30 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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0answers
19 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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0answers
34 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
55 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
34 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
43 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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1answer
43 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 ...