# Questions tagged [valuation-theory]

For questions related to valuation functions on a field, and their corresponding valuation rings.

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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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### Is the dimension of a valuation ring equal to the rank of its value group?

Here it is claimed that: The dimension of a valuation ring is equal to the rank of its value group. I couldn't find confirmation of this statement so I'm trying to prove it myself. Here is what I ...
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### Showing degree of global extension is sum of degree of local extensions

I am trying to understand why, for a global extension $F \subset K$, given a place $\mu$ over $F$, we have that $$n = \sum_{\nu: \nu | \mu} n_{\nu}$$ where $n = [K:F]$, $n_{\nu} = [K_{\nu}|F_{\mu}]$ ...
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### Is a field with archimedean absolute value with compact unit sphere complete?

By the unit sphere of a valued field $(K,|\cdot|)$ I mean $\{x\in K:|x|=1\}$. We know that if the unit closed ball $\{x\in K:|x|\le 1\}$ of a valued field $K$ with nontrivial absolute value is compact,...
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### Completion of global field = extension of completed sub-field?

Given a global field $K = F(\alpha)$ as a finite seperable extension of $F$, where $F = \mathbb{Q}$ or $\mathbb{F}_q(t)$, suppose that $u$ is a fixed place of $F$, and that $v$ is an extension of $u$ ...
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### Is there any relationship between domination of local rings and extension of ring homomorphisms to an algebraically closed field?

What is the relationship between domination of local rings and extension of ring homomorphisms to an algebraically closed field? Let $K$ be a field, $A,B$ local rings contained in $K$. We say $B$ ...
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### Question about Dedekind Domains and Valuations

I've been trying to prove this claim that I think is true, but I'm getting more and more sure it's not true. Here's the set up: Let $\mathcal{O}$ be a Dedekind domain, and $K$ its field of fractions. ...
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### Does a pair of complex embeddings correspond to a single complex prime?

I'm reading through Neukirch's Algebraic Number Theory, and he distinguishes between a number field's real primes, given by its real embeddings, and its complex primes, which are induced by the pairs ...
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### Characterization of Prüfer domains via valuation rings and modules [duplicate]

I am learning valuation rings from Matsumura’s Commutative Ring Theory book. There is an exercise that characterizes Prüfer domains among integral domains. Let $A$ be an integral domain. The exercise ...
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### Is $v((f \circ g)(x)) \geq v(f(x))+v(g(x))$, where $v$ is p-adic valuation?

Let $\mathbb Q_p$ be a $p$-adic field and $f,g \in \mathbb Q_p[x]$. Suppose $v$ is a $p$-adic valuation, and let $f \circ g$ be the formal composition. Is $v(f \circ g) \geq v(f)+v(g)$ ? For the ...
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Let $v:K\to \Gamma$ be a valuation on a field $K$. I know that if the valuation ring $\mathcal O_v$ is henselian and $k$ has characteristic zero, then $k$ has a lift to $K$. I.e. there is a section $s:... • 585 1 vote 0 answers 67 views ### Why's$v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$integer for unramified local field extension$L/K$? If$K$is a local field and$L/K$is a finite extension, then the valuation$v_K$can be extended uniquely to a valuation$v_L$of$L$such that$v_L$restricted to$K$is equal to$v_K$. This is one ... • 5,895 0 votes 1 answer 37 views ### Unramified extension in CDVF Let$(K,v)$a complete discrete valuation field of mixed characteristics$(0,p)$and let$\bar{K}$ann algebraic closure of$K$. We can define an additive valuation$w: \bar{K} \to \mathbb{Q}\cup \{\...
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The classical definition of a field $K$ with an absolute value $|\cdot|:K\to \mathbb{R}_{\geq 0}$ is that $\forall x,y\in K$ $x=0\Leftrightarrow|x|=0$ $|xy|=|x|\cdot|y|$ $|x+y|\leq |x|+|y|$ If the ...