Tagged Questions

For questions related to valuation functions on a field.

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A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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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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Isomorphism of algebras $E_v = \prod_{w\mid v} E_w$

I am currently reading the article Bayerâ€“Fluckiger, Eva, B. Nivedita, and Raman Parimala. "Hasse principle for Gâ€“quadratic forms." Documenta Mathematica 18 (2013): 383-392. in which there is a ...
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Valuation of discriminant

So the discriminant of a polynomial of degree $n$ in the form of determinant of the resultant matrix can be written as $$\det(D)\det(A-BD^{-1}C)$$ where $A, B, C, D$ are block matrices of the ...
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Non-smooth curve in $\mathbb{A}^2$

In one of my exercises on Algebraic Geometry, I showed that the curve $X \subset \mathbb{A}^2$ defined by $x^3-y^2$ is irreducible but not smooth. Furthermore, they ask the following question that I ...
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Valuation fields and linear disjointness

I've got a question about linear disjointness of extension fields. Here I refer to the definition that Wikipedia gives: Two algebras $A$, $B$ over a field $k$ inside some field extension $\Omega$ ...
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Henselization and valued field

What is the importance of henselization in valuation theory, when the rank of valuation is bigger than one? Thanks
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Simple Question about Valuations and Krull Rings

I have what is a very simple question about essential valuations for Krull rings. Before getting to the question, I'll give a sketch of the situation. Any help would be much appreciated. Suppose that ...
$\sqrt[n]{x}$ as a power series in a complete field with absolute value.
Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
Does a place $v$ of a number field $K$ ramify in $L/K$ iff $v\mid d_L$?
Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$. I know that a rational prime $p$ in $\mathbb Q$ ...