For questions related to valuation functions on a field.

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Hensels Lemma in many variables

Let $(K,v)$ be a henselian valued field, with valuation ring $\mathcal{O}$ and residue field $Kv$. Then given a polynomial $f \in \mathcal{O}[x]$, henselianity tells that given some suitable ...
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1answer
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tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :) I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting: $k$ an ...
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Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
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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 ...
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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.?
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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 ...
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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 ...
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1answer
54 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 $x = p^k u v^{-1}$ ...
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Ramification index of infinite primes

I am reading Neukirch's Algebraic Number Theory. On page 184, Chapter 3, Neukirch defines the ramification index of infinite primes as follows: For a finite extension $L/K$ of number fields, and an ...
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23 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) ...
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112 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminates, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by $\nu\left(\sum ...
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1answer
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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 $x$ ...
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1answer
44 views

Extension of a discrete non-archimedean absolute value

Let $P$ be a prime ideal of a Dedekind domain $A$, $v$ an absolute value induced by $P$, and $B$ the integral closure of $A$ in a finite separable extension $E$ of $A$'s quotient field $K$. If $w$ is ...
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1answer
44 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 $\operatorname{Ann}(m) = a \operatorname{Ann}(am)$?
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1answer
50 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 $x\in K$). ...
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1answer
75 views

Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} ...
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1answer
49 views

Formula for Hilbert symbol for primes lying above 2

Let $K$ be any number field, and let $\mathfrak{p}$ be a prime lying over 2. Is there a formula for computing the Hilbert symbol $(a,b)_\mathfrak{p}$? I know the formula when $\mathfrak{p}$ lies above ...
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67 views

Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
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1answer
75 views

Lattice in a vector space of dim 2 over a valuated field.

I'm reading "Arbres, amalgames et SL2" of J.P. Serre, and something is not clear to me, but is to him :) Let $k$ be a field, with a discrete valuation $v$, ie a group epimorphism $v:k^\ast \to ...
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1answer
32 views

Non-archimedean valuation

In the definition of non-archimedean valuation the value group is the group of real numbers $\mathbb R$. Can we replace $\mathbb R$ by any totally ordered group different from $\mathbb R$ ? Thanks
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1answer
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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 ...
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1answer
55 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 ...
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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
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164 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 ...
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1answer
64 views

How to show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$?

How can I show that $|z|=(z\overline{z})^{1/2}$ is the only valuation of $\mathbb{C}$ that extends the absolute value of $\mathbb{R}$? I can see that it suffices to work with the unit disk, i.e. it ...
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1answer
60 views

Localizations of the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$

Consider the ring $A=\Bbb{C}[X,Y]/(Y^2-X^4+2X^2-1)$. Let $A_M$ denote the localization of $A$ with respect to maximal ideal $M$. My question is: Is $A_N$ a DVR, where $N$ is the maximal ideal ...
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0answers
66 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}) ...
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1answer
164 views

Definition of algebraic variety

In general, the definition of an algebraic variety differs from one reference to other. The definition that I was used to is to consider an algebraic variety as an integral scheme of finite type. ...
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1answer
108 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 ...
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2answers
152 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 ...
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1answer
88 views

Embedding $\mathbb Q^c$ into $\mathbb Q^c_p$

Let $p$ be a prime number and $\mathbb Q_p$ the $p$-adic completion of $\mathbb Q$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. Is there an embedding $$j: \mathbb Q^c ...
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103 views

Question about complete DVR.

Is there a simple proof that you know to the following statement: If the residue field $k$ of a complete DVR $R$ has the same characteristic as $R$, then $R$ contains a subfield isomorphic to $k$. ...
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1answer
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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 ...
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1answer
136 views

Valuation Ring with value group $\Gamma=\mathbb{Z} \oplus \mathbb{Z}$

Let $\Gamma=\mathbb{Z}\oplus \mathbb{Z}$ be the free abelian group with two generators, lexicographically ordered. That is, $(a,b)\geq (a^{'},b^{'})$ iff either $a>a^{'}$ or $a=a^{'} \mathrm{and}\, ...
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1answer
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Discrete valuation rings are infinite.

Assume $F\supset k$ is a functional field. Assume $R\subset F$ is a discrete valuation ring with a quotient filed $F$, that contains the field of constants $k$. Assume $t$ is a local parameter for ...
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1answer
89 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 ...
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Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayer–Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for G–quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
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1answer
131 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 ...
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2answers
116 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 ...
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1answer
34 views

A certain ideal of a valuation ring

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
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Two discrete valuations on a Dedekind domain

A question from Frohlich and Taylor's book 'Algebraic Number Theory', page 60. Let $\mathfrak o$ be a Dedekind domain with quotient field $K$ and let $v$ be a discrete valuation on $K$. Let ...
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How to prove that a DVR is not complete

My question is inspired by a comment in this topic. How to prove that $R=\mathbb C[x]_{(x)}$ is not complete in the topology of its maximal ideal? One knows that $R$ is a DVR, and its field of ...
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2answers
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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 ...
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1answer
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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 ...
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1answer
39 views

How to show that $R_{\mathfrak{m}}$ is $R$?

Let $R$ be a discrete valuation ring and $\mathfrak{m}$ its unique non-zero maximal ideal. How to show that $R_{\mathfrak{m}}$ is $R$ using definition of a discrete valuation ring? I know that ...
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Question about localizations of discrete valuation rings.

Let $R$ be a discrete valuation ring. Then $R$ has only two prime ideals: $0$ and the maximal ideal $\mathfrak{m}$. It is said in Hartshorne, page 74, Example 2.3.2 that the localization of $R$ at $0$ ...
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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 ...
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2answers
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Primes corresponding to embeddings of a number field

Let $k$ be a number field. Define a prime of $k$ to be an equivalence class of absolute values on $k$. If $\sigma:k\hookrightarrow \mathbb{C}$ is an embedding of $k$ into the complex numbers then we ...
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1answer
64 views

Integral elements of a non-archimedean completion of a number field

Let $k$ be a number field, $v$ a discrete non-archimedean valuation on $k$. Let $k_v$ be the completion of $k$ wrt $v$ and $\mathcal O_v$ its valuation ring. My question is: If $x\in k_v$ then is ...
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1answer
35 views

Module Notation $Dx$

Let $K$ be a field, and let $D$ be a subring of $K$ with identity. Let $K^*$ be the multiplicative group of nonzero elements of $K$. The group $U$ of units of $D$ is a subgroup of $K^*$. We take ...