# Questions tagged [valuation-theory]

For questions related to valuation functions on a field.

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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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### Normal basis of finite extension of a complete DVR

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?...
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### Valuation of a number field element over a prime ideal in an order

Having read https://mathoverflow.net/questions/144671/number-field-sieve-for-factorization-with-non-monic-non-linear-polynomial-cant I stumbled on a problem I can't prove. Most of the questions posed ...