# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

115 questions
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### An infinite prime can ramify right? (So what is Neukirch talking about?)

I have been under the impression for several years that if $L/K$ is an extension of number fields, then an infinite place of $K$ is said to ramify in $L$ if it comes from a real embedding of $K$ which ...
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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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### question on proof in Frohlich/Taylor regarding extension of a discrete absolute value on a complete field

I am having some trouble understanding a proof in Fröhlich and Taylor, pages 105-106. There $K$ is a complete field with a discrete absolute value $u$ $\mathfrak o,\mathfrak p$ is the valuation ring, ...
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### Projective general linear group on discrete valuation ring

Let $R$ be a complete discrete valuation ring and $k$ its residue field. Let $H$ be a finite subgroup of $PGL_2(k)$ such that its order is prime with char($k$). Is there some elementary way to show ...
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Let $K$ be a field. Let $\mathcal{O}_K$ be the intersection of all valuation rings with quotient field $K$. Can someone give an example of a field $K$ in which we don't have a bijection of sets: $$\... 0answers 99 views ### Computing a valuation of a field Assume k is an algebraically closed field, and x and y are transcendental over k. I want to compute the valuation ring of F, the field of fractions of the ring A=k[x,y]/I, where I=\langle ... 0answers 123 views ### Equivalent Hensel's lemma? Let K be a complete field with nonarchimedian valuation |\cdot| and \mathcal{O}_K := \{x \in K: \; |x| \le 1\}, \mathfrak{m} := \{x \in K: \; |x| < 1\}. I have seen two statements that do ... 0answers 69 views ### A question on a sum of valuations Let A be a discrete valuation ring of characteristic zero. Let v be the valuation on A. Let I be a finite index set and d_i a positive integer for all i in I and define$$ d:= \sum_{i \...
Let $R$ be a complete discrete valuation ring, and $S$ be a finite extension such that the associated residual field extension is separable. Then, why is it possible to choose a normal basis in powers?...