For questions related to valuation functions on a field.

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5
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3answers
391 views

Examples of Non-Noetherian Valuation 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 ...
0
votes
1answer
172 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) ...
9
votes
4answers
445 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 ...
3
votes
2answers
428 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 ...
5
votes
3answers
672 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 ...
7
votes
1answer
124 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
4
votes
1answer
151 views

A question on valuation overrings of a PID

Let $A$ be a PID and let $K$ be its quotient field. Let $V$ be a valuation ring of $K$ containing $A$ and assume $V\neq K$. Show that $V$ is a local ring $A_{(p)}$ for some prime element $p$. I ...
2
votes
2answers
95 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 ...
2
votes
2answers
69 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\{ ...
2
votes
1answer
172 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) = ...
7
votes
2answers
190 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 ...
4
votes
0answers
138 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, ...
3
votes
1answer
41 views

Valuation rings and total order

Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). ...
0
votes
1answer
74 views

Discrete valuations of a functional field have discrete valuation rings.

Theorem: If $\nu:F\to\mathbb R\cup\{\infty\}$ is a valuation of a functional field, then the set $$\mathfrak O_{\nu}=\{x\in F: \nu(x)\geq 0\}$$ is a local ring with maximal ideal $$\mathfrak ...
0
votes
0answers
54 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 ...