For questions related to valuation functions on a field.

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38 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\...
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0answers
61 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}, x\...
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2answers
42 views

Is every integral domain contained in a discrete valuation ring?

Is is true that every integral domain which is not a field is contained in a proper subring of its fraction field which is a DVR?
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26 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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1answer
29 views

Ramification group definition in Neukirch's Chap. 2 Section 9

Let $L/K$ be an algebraic Galois extension and $w/v$ a non-archimedean extension of valuations. Let $\mathcal{O}$ and $\mathfrak{P}$ denote the valuation ring and valuation ideal of $(L,w)$. After ...
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1answer
34 views

Property of valuation in $\Bbb{Z}_p$

Let $v$ denote the $p$-adic valuation in $\Bbb{Z}_p$. Let $a_1,a_2,a_3$ be three elements of $\Bbb{Z}_p$. Then I want to show that $$\min \{ v(a_1), v(a_2), v(a_1+a_2+a_3)\}=\min \{v(a_1),v(a_2),v(...
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1answer
30 views

Topology for the decomposition of $L\otimes_K K_\frak p$

Let $L/K$ be an extension of number fields, $\frak P|\frak p$ primes of $\mathcal O_L$ and $\mathcal O_K$, and let $K_\frak p$ be the completion of $K$ wrt $|\cdot|_\frak p$ and $L_\frak P$ the ...
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2answers
47 views

Why does discrete valuation need to be surjective?

My question appeared while I was solving a problem in Abstract Algebra of Dummit and Foote. That is problem 26. My solution is almost the same as the proof in this link. However, I'm still ...
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2answers
34 views

finite extensions of discrete valuation fields: A method to find a basis

Suppose that $L|K$ is a finite extension of discrete valuation fields. Namely $w$ is a discrete valuation on $L$ extending a valuation $v$ on $K$. Now consider the respective rings of integers $\...
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2answers
68 views

Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
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2answers
55 views

Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
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1answer
23 views

When do we have equality in the third axiom of valuations?

Let $R$ be a ring and $v: R \to \mathbb Z \cup \{+\infty\}$ a map that meets following conditions: $v(a) = +\infty \iff a=0$ $v(ab) = v(a)+v(b)$ $v(a+b) \geq \min\{v(a),v(b)\}$ I have to show that ...
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1answer
19 views

An extension $w$ of a valuation $v$ induced from $\mathfrak{p}$ come from ideal $\mathfrak{q}$ above $\mathfrak{p}$?

Assume the standard context for extension of valuations. An extension $w$ of a valuation $v$ induced from a prime ideal $\mathfrak{p}$ comes from a prime ideal $\mathfrak{q}$ above $\mathfrak{p}$? ...
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0answers
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) = \...
1
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1answer
41 views

Separable extensions of complete valued fields are automatically totally ramified?

I come up with the following argument which seems to be too good to be true: Suppose that $L|K$ is finite separable extension of complete valued fields. Let $\nu, \nu'$ be the valuation on $L$ and $K$...
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1answer
31 views

Separability of complete value fields and residue class fields

Let $L|K$ be a finite separable extension of fields complete under some valuation and let $\lambda, \kappa$ be residue class fields of $L$ and $K$ respectively. I guess that we do not know $\lambda|\...
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0answers
58 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\...
1
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1answer
34 views

Relationship between residue class fields between extension

Let $K$ be a field with respect to a valuation and $L$ be a finite extension. For simplicity, assume $K$ is complete. From theory, valuation on $K$ extends to $L$ and the extended valuation on $L$ is ...
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1answer
15 views

Proving that certain limits exist

Let $K$ be a discrete valuation field where $\nu:K\longrightarrow\mathbb Z$ is a surjective valuation. Let $\lambda\in]0,1[\subseteq\mathbb R $, then the valuation $\nu$ induces a metric $d_\nu$ on $K$...
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0answers
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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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 $K$...
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0answers
108 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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1answer
27 views

Extending valuations and linear disjointness of fields

Let $F$ be a field and let $K$ be a field extension of $F$. Suppose that $K$ can be written as the compositum of field extensions $E$ and $L$ of $F$, linearily disjoint over $F$, thus $K$ can be ...
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2answers
67 views

Reduction map on torsion points of an elliptic curve and their valuation

Let $K$ be a field of characteristic zero, complete with respect a discrete valuation $v$. Assume that the residue field $k$ is of positive characteristic $p$. Now take an elliptic curve $E$ defined ...
2
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1answer
52 views

Quotients in a non-discrete valuation ring

Let $R$ be a valuation ring, $\mathfrak{m}$ the maximal ideal of $R$. Let $k$ be the residue field, and $K$ the field of fractions of $R$. Assume that the valuation on $K$ is such that $v(K)=\mathbb{Q}...
3
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1answer
82 views

roots of unity in the maximal unramified extension and Kummer extensions

Let $K$ be a field complete with respect to a discrete valuation $v$ with residue field $k$ of positive characteristic $p$. Consider $$ K \subset K_{ur} \subset K_s$$ with $K_s$ the separable ...
2
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1answer
66 views

Associate a discrete valuation ring to a field $k$.

I have a field $k$ of positive characteristic $p$, not necessarily perfect. Can i find a discrete valuation ring that have $k$ as residue field and field of fractions $K$ of characteristic zero? I ...
4
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0answers
48 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
25 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
27 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) \...
2
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1answer
45 views

Two dimensional valuation domain with value group $\Bbb Z \oplus \Bbb Q$

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 $P^2 \neq P$. Can we say $P^n \neq P^{n+1}$ for ...
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0answers
52 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 ...
3
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0answers
73 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
105 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
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
40 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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1answer
33 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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0answers
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 $R$ ...
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3answers
52 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
63 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
111 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 $\...
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1answer
52 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 ...
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0answers
146 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
93 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
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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1answer
36 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 $n\geq 1$. ...
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2answers
197 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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votes
2answers
78 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 non-...
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1answer
49 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 $(\hat{K},\hat{v})...
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1answer
42 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 $...