Questions tagged [valuation-theory]
For questions related to valuation functions on a field, and their corresponding valuation rings.
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Valuation in the local ring of a curve
Let $k$ be an algebraically closed field and $A = k[X,Y]/(Y^2-X^3)$. Let $M$ be the ideal of $A$ generated by $X$, $Y$, and $A_M$ be the localization of $A$ at $M$.
Is $A_M$ a discrete valuation ring?...
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Lifting subextension of residue fields
Let $F \subset L$ be a arbitrary extension of valued fields with associated extension of residue fields $k \subset l$. Suppose we are given a subextension $k\subset k'\subset l$ can we always find an ...
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Tamely ramified extensions of $K_{\mathfrak p}^{\mathrm{unr}}$
I have some doubts regarding this statement, which I don't know if it's true:
Statement:
Let $K_{\mathfrak p}$ be the completion of a number field w.r.t. the $\mathfrak p$-adic valuation, and let $K_0$...
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Extending the “point at infinity” valuation for function fields
I understand that if $A$ is a Dedekind domain and $K=Frac(A), L/K$ finite separable, and $B$ is integral closure in $A$, then $\mathfrak{p}$ is a prime then there is a bijection between places ...
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$K^{ab}=K^{ur}M$, $M$ maximal, abelian, totally ramified
I read this property $K^{ab}=K^{ur}M$ where $M$ is a maximal totally
ramified abelian extension of $K$ (local field) as a corollary of the
following
Let $L/K$ abelian extension of a local field $K$. ...
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Example of a complete valued field that does not satisfy hensel's lemma
A valued field $(K,v)$ is henselian if it satisfies hensel's lemma.
Notation : $O_v$ is the valuation ring of $(K,v)$ (it is a local ring), $F_v$ is the residue field of $F_v$.
Hensel's Lemma : for ...
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Using Nakayama's lemma in non-local ring
Let $R$ be a Noetherian integral domain of dimension one and $\mathfrak{m}$ an ideal such that $\text{dim }\mathfrak{m}/\mathfrak{m}^2=1$ as an $R/\mathfrak{m}$-vector space. The localization of $R$ ...
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Uniformizers in one-dimensional local rings
Suppose that $R$ is a Noetherian ring with unique maximal ideal $\mathfrak{m}$. Further suppose that $\mathfrak{m}/\mathfrak{m}^2$ is a one-dimensional $R/\mathfrak{m}$-vector space. By an application ...
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Set of representatives vs Teichmüller representatives
Say $K$ is a local field (complete wrt a discrete valuation with finite residue field, or if you want perfect residue field). Denote its residue field by $k$ where $k= \mathcal{O}_K / \mathcal{M}_K.$ ...
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What's the importance of the Approximation Theorem - Artin-Whaples Approximation Theorem
The above pictures present the statement of the approximation theorem by Artin-Whaples and its corollary in their paper. I do understand that the theorem implies that we can use one element to ...
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What is the connection between the valuation of a polynomial of a function filed of a curve and the corresponding laurent series?
I've read somewhere an MSE that we can understand the normalized valuation of a polynomial in the function field of a curve at a smooth point as the first non-vanishing coefficient (or exponent) of a ...
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Why is (y) a uniformizer for $y^2 = x^3 + x$ at (0,0) and why is its order equal to 1?
I'm currently trying to understand Silvermans example for the valuation on curves discussed in the answer to this post: Definition and example of "order of a function at a point of a curve"
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Why is the valuation of a separable closure equal to $\mathbb{Q}$?
I am reading a paper by Serre, in which at some point he say
We extend the valuation $v$ from $K$ to $K^{sep}$; in this way we obtain a valuation on $K^{sep}$ with value group $v(K^{sep \times})=\...
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Convergence of $\left \{c^{q^k}\right \}$ with finite residue field with $q$ elements
The problem
Let $ν$ be a normalised discrete order function of a field $F$. Suppose that $F$ is complete. Put
$$R=\left\{x\in F:v\left(x\right)\geqslant0\right\}$$
and
$$ M=\left\{x\in F:v\left(x\...
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Prove the p-adic valuation of n! equals a floor function sum
I've seen related questions, but none quite like this one. For a rational number $r$ let $[r]$ be the largest integer less than or equal to $r$, e.g., $[\frac{1}{2}] = 0, [2] = 2$, and $[3\frac{1}{3}] ...
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