For questions related to valuation functions on a field.
3
votes
1answer
44 views
Discrete Valuation Ring and Subring of the Fractions Field
Let $R$ be a Discrete Valuation Ring, and $K$ its fractions field. Now if $B\subseteq K$ is a subring with $R\subseteq B$ then we have $$B=R \text{ or } B=K.$$
Now this seems to be a very basic ...
0
votes
2answers
75 views
Valuation but not Noetherian Rings
For valuation rings I know examples which are Noetherian.
I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind?
I am very ...
3
votes
0answers
59 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, ...
2
votes
2answers
80 views
The composite of all unramified extensions inside an algebraic closure
I'm reading Ch.II, $\S$ 7 of Neukirch's Algebraic Number Theory and I'd be really grateful if someone could help me understand the following:
Let $K$ be a complete valued field wrt a non-archimedean ...
2
votes
2answers
36 views
Valuations, Isomorphism, Local ring
Let $x \in \mathbb{Q}, \, x \neq 0$, such that $x=p^r \frac{a}{b}, \quad a,b, r \in \mathbb{Z}, \quad p \nmid a, \quad p\nmid b$.
Let $v_p(x):=r$ and $v_p(0):= \infty$. Also,
$$\mathcal O_p= \left\{ ...
-1
votes
1answer
112 views
Discrete Valuation Rings problem 2
An order function on a field $K$ is a function $\phi:K\to \mathbb{Z} \cup {\{\infty}\}$ satisfying:
i) $\phi(a) = \infty$ if and only if $a=0$.
ii) $\phi(ab) = \phi(a) + \phi(b)$.
iii) ...
0
votes
0answers
27 views
A trace formula for an extension of complete valued fields
I'm trying to understand the following fact.
Proposition. Let $L/K$ be a finite separable extension of complete valued fields, with respective valuations $w$ and $v$, valuation rings $\mathcal{O}_L$ ...
2
votes
0answers
39 views
A question on a sum of valuations
Let $A$ be a discrete valuation ring of characteristic zero. Let $v$ be the valuation on $A$.
Let $I$ be a finite index set and $d_i$ a positive integer for all $i$ in $I$ and define $$ d:= \sum_{i ...
4
votes
1answer
63 views
About Henselization
I have some question about Henselization of valued field. If $(K_{1}, \nu_{1})$ is a Henselization of valued field $(K, \nu)$. Which one is true.
$K_{1}/K$ is an algebraic extension.
$K_{1}/K$ is a ...
2
votes
2answers
43 views
Valuation rings and domination
I was wondering if the following holds: if $A$ is a valuation ring with maximal ideal $m_A$ and $B$ is a ring extension of $A$ contained in $Frac(A)$ then the only maximal ideal of $B$ is still $m_A$.
...
1
vote
1answer
60 views
Properties of valuation map
It seems not so hard to prove but how can we prove by induction. Let $K$ be a field and $\nu$ be a valuation map. If $a_{1} + a_{2} + ... + a_{n} = 0$ then prove that $\nu(a_{i}) = \nu(a_{j})$ for ...
1
vote
1answer
74 views
A question in valuation theory
Let $R$ be an integral domain with quotient field $K$. Let $P$ be a prime ideal of $R$, and $L$ be a transcendental extension of $K$. I think there is more than one valuation domain $O'$ with quotient ...
2
votes
1answer
83 views
Construction of a Valuation Ring via Zorn's Lemma, except not
In Atiyah & MacDonald they provide an abstract way to construct valuation rings, I'm curious how easy it is to work this out in general. As a refresher let $K$ be a field, $L$ an algebraically ...
0
votes
2answers
57 views
Function field has infinitely many valuations.
Suppose we have a field $K$ which is a finite algebraic extension of the field $\mathbb{C}(X)$. Can you give me an argument that $K$ admits infinitly many discrete valuations?
6
votes
2answers
167 views
Non-trivial valuation of $\mathbb R$
In a valued fields book on page $82$ there is a question: "show that every non-trivial valuation of $\mathbb R$ has divisible value group and algebraically closed residue class field." How do I ...
4
votes
1answer
108 views
Ring of integers in a field of fractions
Let $R$ be ring with complete non archimedian absolute value.
Let $Q$ be the associated field of fractions with the extended absolute value.
Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
3
votes
1answer
69 views
Real valuations on Dedekind domains
Let $D$ be a Dedekind domain. Let $v:D \to \mathbb{R}$ a valuation.
We know that for every prime ideal $\mathfrak p$ of $D$ the localization $D_{\mathfrak p}$ is DVR.
Does every valuation on $D$ ...
4
votes
1answer
66 views
Henselization and immediate extension
I am reading about the henselization and immediate extension of valuation. I am getting confusion about some basic terminology. I have few question. \ $1)$ Is every hensilization extension of ...
2
votes
1answer
77 views
Valuation over the algebraically closed field of rational number
How do we define the valuation over the algebraically closed field of rational numbers say $\bar{\mathbb Q}$ as an extension of the valuation of $\mathbb Q$ ?
4
votes
1answer
397 views
Extension of valuation
We define a valuation on the field of rational number $\mathbb Q$ as follows. For example if we choose a prime number $2$ then for $x \neq 0\in \mathbb Q$, $v(x) = v(2^{n}a/b)= n$ where $n$ is an ...
1
vote
1answer
64 views
Valuation rings of complete non-archimedean fields which are not local
I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
1
vote
0answers
54 views
Henselization and valued field
What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
3
votes
2answers
84 views
What is a “normalized valuation” corresponding to a valuation ring?
I encountered the phrase "normalized valuation" similar to the following:
Let $A_i$ be the valuation ring $k[x_1,...,x_n]_{\langle x_i\rangle}$ and $v_i$ be the normalized valuation defined by ...
6
votes
2answers
128 views
How many absolute values are there?
My question is the following: Are there algebraic norms on the fields $\mathbb{R}, \mathbf{Q_p}$ ''other'' than the absolute value, respectively $|\cdot|_p$?
Now phrasing more precisely: If generally ...
1
vote
0answers
130 views
Non-Archimedean Fields
Are there non-Archimedean fields without associated valuation or being a non-archimedan field implies it is a valuation field?
I understand that a non-Archimedean field is a field which does not ...
0
votes
1answer
139 views
What is a valuation associated to an ordering on a field?
If $(K,\leq)$ is a totally ordered field with $P\!=\!\{\alpha\!\in\!K;\, 0\!\leq\!\alpha\}$, how is the valuation associated to $P$ defined?
I was searching through Prestel & Delzell's Positive ...
3
votes
1answer
88 views
A property of non-Archimedean metrics
I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers,
$p$-adic Analysis, and Zeta-Functions), and came across the exercise:
Prove that a norm $\|.\|$ on ...
9
votes
4answers
265 views
Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?
I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this:
Show that $x^2-82y^2=\pm2$ has solutions in every ...
0
votes
1answer
84 views
Ramification of local field
Let $K$ be finite field, $E=K((x))$ and $F=K((x^n))$, that is $F$ is field of formal laurent series in $x^n$. I know that $E\cong F$ (because you can consider $x\rightarrow x^n$ )
I want to prove if ...
4
votes
0answers
102 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 ...
2
votes
1answer
152 views
Existence of an element of given orders at finitely many prime ideals of a Dedekind domain
Let $A$ be a Dedekind domain.
Let $P$ be a non-zero prime ideal of $A$.
Let $\alpha \in A$.
Let $k$ be a non-negative integer.
If $\alpha \in P^k$ and $\alpha\notin P^{k+1}$, we write $v_P(\alpha) = ...
2
votes
1answer
68 views
Is a valuation a continuous map?
What I want to know is: the preimage of an integer by a valuation map is an open set?
Otherwise:
Can we cover a valuation field by open sets with elements with fixed valuation?
2
votes
2answers
148 views
Why is the restricted direct product topology on the idele group stronger than the topology induced by the adele group?
I am confused by the saying that the restricted direct product topology on the idele group is stronger than the topology induced by the adele group. And perhaps this is elaborated by the following ...
3
votes
0answers
60 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}, ...
6
votes
1answer
162 views
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.
The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$.
I was trying ...
2
votes
2answers
120 views
Problem in valuation theory
Find $\alpha\in \mathbb{Q}$, such that $v_2(\alpha-1/3)\ge 2$, $v_3(\alpha-1/2)\ge 3$ and $|\alpha-1|_\infty<1/2$, where $v_p$ is the $p$-adic exponential valuation and $|\cdot|_\infty$ is the ...
2
votes
1answer
72 views
Difficulty with nonarchimedean valuation exercise
I'm reading through a book on algebraic number theory, and in the opening section on valuation theory I've been stumped by one of the problems. The problem is in two parts; the first says
"Let ...
0
votes
0answers
46 views
What does it mean for a valuation to be normed?
I have a homework problem that uses the term: "a discrete normed non-trivial valuation" on a field.
We've defined the discrete trivial valuation in class, so that part is clear.
I think the natural ...
1
vote
0answers
116 views
units in discrete valuation rings
Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to ...
5
votes
1answer
98 views
Discrete valuation ring extension such that $A[\pi]$ is not integrally closed
Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero.
Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring?
If not, ...
0
votes
1answer
41 views
How do I compute the discrete valuation of the sum of two elements
Let $O$ be a discrete valuation ring with valuation $v$. We normalize by $v(\pi) =1$, with $\pi$ a prime element in $O$.
By definition, for all $x,y$ in $O$, we have $v(x+y) \geq \min (v(x),v(y))$. ...
0
votes
1answer
85 views
Algebraically maximal valued fields
Does anyone know of an elementary proof that an algebraically maximal field is Henselian (ie one that does not assume knowledge of henselizations)?
Definitions:
We say a valued field $(K,v)$ is ...
4
votes
2answers
404 views
Roots of unity in a local field
The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as
$K=\langle \pi\rangle\times \mu_{q-1}\times ...
4
votes
2answers
142 views
Algebraic Henselian extensions of the rationals
Let $p$ be a fixed prime, $v:\mathbb{Q}\rightarrow\mathbb{Z}$ be the $p$-adic valuation on $\mathbb{Q}$ and $\mathbb{Q}^h$ the Henselization of $\mathbb{Q}$ with respect to $v$. I want to show that ...
0
votes
1answer
89 views
A construction in the proof of “ any local ring is dominated by a DVR”
Let $O$ be a noetherian, local domain with maximal ideal $m$. I want to prove: for a suitable choice of generators $x_1,\dots,x_n$ of $m$, the ideal $(x_1)$ in $O'=O[x_2/x_1,\dots,x_n/x_1]$ is not ...
1
vote
1answer
112 views
Bounding $p$-adic valuations in inequality
I'm developing an algorithm that comes across inequalities of the form
\begin{align*}
\operatorname{ord}_p(c(b)) > \alpha
\end{align*}
for some polynomial $c \in \mathbb{Q}[b]$, $c(b) = c_0 + c_1b + ...
1
vote
2answers
189 views
A characterisation of tame ramification
The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155)
Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
5
votes
3answers
367 views
Algebraic Closure of Puiseux Series
Using the construction $R_N = K[t^\frac1N]$ $L_N = Quot(R_N)$ and $P = \bigcup_{N\in \mathbb{N}} L_N$ one automatically gets that the puiseux series are a field. Nevertheless they are also an ...
13
votes
1answer
306 views
Why is $\mathbb{C}_p$ isomorphic to $\mathbb{C}$?
I know that two closed fields of caracteristic $0$ and uncountable are isomorphic iff they have the same cardinality. But I don't know why $\mathbb{C}_p$ has the same cardinality as $\mathbb{C}$. Can ...
3
votes
1answer
79 views
Dividing the ramification index of an extension
Let $f(x)$ be a polynomial of degree $m$ over $\mathbb{Q}_{p}$ with all roots $r_{i}$ such that $\operatorname{ord}_{p} r_{i} = \frac{1}{p}$. Why does $p$ have to divide the ramification index of ...
