0
votes
1answer
17 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 $\alpha \in B$ ...
1
vote
1answer
21 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
votes
1answer
34 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 ...
4
votes
1answer
86 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
votes
0answers
30 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.?
1
vote
1answer
84 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
votes
0answers
38 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
votes
1answer
42 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 $x = p^k u v^{-1}$ ...
0
votes
0answers
34 views

Showing that a set is Discrete Valuation Ring

An order function on a field $K$ is a function $ϕ:K→Z∪{∞}$ satisfying: i) $ϕ(a)=∞$ if and only if $a=0$. ii) $ϕ(ab)=ϕ(a)+ϕ(b)$. iii) $ϕ(a+b)≥min(ϕ(a),ϕ(b))$. Show that R=$\{z∈K∣ϕ(z)≥0\}$ is a DVR ...
0
votes
0answers
18 views

A valued field is complete iff its ring of intergers is complete

Let $K$ be a field, and let $v$ be a valuation defined on K, and let $O$ be the ring of integers, and $M$ be the maximal ideal. How to show that $K$ is complete (i.e $K$ is equal to its completion) ...
2
votes
1answer
26 views

Valuation rings (confusion with arithmetics)

I am reading Goldschmidt's Algebraic functions and projective curves. From the book: Let $K$ be a field. An integral domain $\mathcal{O}\subset K$ is a valuation ring if for all $x\in K$ either $x$ ...
0
votes
1answer
36 views

Annihilators in discrete valuation rings

Let $A$ be a discrete valuation ring and $M$ be an $A$-module. Let $a \in A$ and $m \in M$ such that $am \neq 0$. Is it true that $\operatorname{Ann}(m) = a \operatorname{Ann}(am)$?
3
votes
1answer
34 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 $x\in K$). ...
3
votes
1answer
53 views

Integral Closure in an Unramified Extension is Generated by a Single Element

Let $R$ be a discrete valuation ring with quotient field $K$, and $L/K$ a finite separable extension which is unramified over $K$. Also suppose that $K$ is complete with respect to the valuation of ...
1
vote
1answer
39 views

Subring of a field with a discrete valuation is a euclidean domain

I am confused by the following problem in Aluffi's Algebra Chapter Zero... A discrete valuation on a field $k$ is a surjective group homomorphism $v : k^* \to (\Bbb Z,+)$ such that $v(a + b) \geq ...
0
votes
1answer
42 views

Help in this proof on DVRs

I'm trying to understand this proof: Anyone could clarify the converse please? I really need help. Thanks a lot
4
votes
1answer
141 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 ...
0
votes
0answers
59 views

Application of Hensel's lemma

Show that the polynomial $\Phi(x)=x^2 -2 \in O(\widehat{\Bbb Q_2})[x] $ has no root in $\widehat{\Bbb Q_2}$, even though $\bar\Phi(x)\in E(\widehat{\Bbb Q_2})[x]$ has a root in $E(\widehat{\Bbb Q_2}) ...
0
votes
1answer
96 views

Non-archimedean valuation on a field

It was a exercise that one of our professors gave it to us and I don't know the solution of it: Suppose that $v$ is a non-archimedean valuation on the field $F$ and $o(v)$ is the valuation ring of ...
3
votes
2answers
111 views

Finding a Uniformizer of a Discrete Valuation Ring

Suppose I have a discrete valuation ring. Then what are some techniques for explicitly finding a uniformizer? I'm especially interested in situations where the ring is given similarly to the following ...
4
votes
1answer
42 views

Discrete valuation on a field - equivalent statements

I have a question and I am stuck, although it should not be too difficult. We consider $K$ a field, $v$ a discrete valuation on $K$ and $O=\{x \in K:v(x)\geq 0\}$ the valuation ring of $v$. Let ...
7
votes
1answer
121 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 ...
2
votes
2answers
115 views

Proving that a discrete valuation-like function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ is a $p$-adic valuation

This problem is from Birkhoff and Maclane, A Survey of Modern Algebra, pg 21, problem 4*. Given a function $w: \mathbb{Z}\backslash\{0\} \rightarrow \mathbb{N}$ that behaves like a discrete ...
1
vote
2answers
41 views

Show $m_v$ is maximal in $R_v$, valuation ring ($v:K\rightarrow G\cup\{\infty\}$ a valuation on $K$)

Let $K$ be a field, $G$ a totally ordered group and $v:K \rightarrow G\cup\{\infty\}$ be a valuation on $K$. I am trying to show that $m_v=\{k \in K: v(k)>0\}$ is a (the?) maximal ideal in ...
1
vote
1answer
42 views

Simple property of a valuation on a field

Let $F$ be a field and $v:F\rightarrow G\cup\{\infty\}$ be a valuation on $F$ so $G$ is a totally ordered abelian group with $\infty$ having the properties $\infty+\infty=g+\infty=\infty+g=\infty$ and ...
3
votes
1answer
58 views

Regarding valuation on function field

Notations: Let $k$ be a field of characteristic $\neq 2$, $k(X)$ be a function field. Suppose $p(X)\in k[X]$ is an irreducible polynomial and $\alpha$ be a root of $p(X)$. Let ...
0
votes
1answer
67 views

discrete valuation ring

I am struggling to understand the proof of the following proposition Let $A=\{x\in K|v(x)\ge 0\}$ for a field $K$ be a discrete valuation ring. Let $t\in A$ s.t. $v(t)=1$. Then any element $x\in A$ ...
3
votes
2answers
130 views

Learn about valuations, valuation rings, value group

I am reading a paper for a summer research project (Example of an interpolation domain ). I am unfamiliar with some of the terms used here and I have tried searching on google for definitions but I am ...
6
votes
1answer
106 views

Valuations on a field and ramification

For $K\subseteq L$, where $L$ is a finite number field extension of $K$, we consider $p\subset R_K$ and $p'\subset R_L$ where $p'$ lies over $p$, where $R_K$ is the ring of integers of $K$ and $R_L$ ...
1
vote
1answer
184 views

An application of the Weak Approximation theorem - Artin-Whaples Approximation Theorem

Let us recall the weak approximation theorem from Valuation theory in Algebraic Number Theory. Let K be a field, and let $|\cdot|_{1},\cdots, |\cdot|_n$ be pairwise non-equivalent nontrivial ...
3
votes
2answers
220 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 ...
5
votes
3answers
567 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
336 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 ...
9
votes
2answers
303 views

Concrete examples of valuation rings of rank two.

Let $A$ be a valuation ring of rank two. Then $A$ gives an example of a commutative ring such that $\mathrm{Spec}(A)$ is a noetherian topological space, but $A$ is non-noetherian. (Indeed, otherwise ...