I am looking for the original reference for Ostrowski's theorem of 1916 that the only valuations on the rational numbers are the trivial, Archimedean, and p-adic valuations. http: …

I'm working through Local Fields by Serre and am stumped by something that he thinks should be obvious. Let $A$ Be a complete D.V.R with uniformizer $\pi$ and $\overline{K}$ be i …

Define asymptotic as a class of sequences {$ x_i$},$_{i\in\mathbb N}$ modulo equivalence {$x_i$}={$y_i$} if $\lim_{i\to\infty} (x_i/y_i)=c\in\mathbb R,c\ne 0$. More, we define $X= …

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is …

This is a similar question from the book "Valued Fields by Antonio J. Engler and Alexander Prestel, Springer, 2005 " page 82, Exercise 3.5.4.(b). Let $(K_{1}, V_{1})\subseteq (K_{ …

Let $(K, \nu)$ be a valued field and $x$ is transcendental over $K$. Is there exist a henselian extension of $(K, \nu)$ in between $(K, \nu)$ and $(K(x), \nu^{'})$ where $\nu^{'}$ …

Let $(k, \nu)$ be a valued field $(k_{1}, \nu_{1})$ is a Hensilization of $(k, \nu)$. Is there exist a another valuation $\nu^{'}$ on $k$ different from $\nu$ such that $(k_{1}, \n …

Let $K$ be a valued field. We say that $K$ is algebraic maximal if any algebraic extension of $K$ has either a bigger value group or a bigger residue field. Under which condition i …

A valued field is said to be algebraic maximal if all its algebraic extension have either a bigger value group or a bigger residue field. Do you know for which field this is a fir …

Let $A$ be a commutative ring, $B$ (resp. $C$) be a commutative $A$-algebra endowed with a valuation $v$ (resp. $w$), not necessarily of rank 1. Assume that $v$ and $w$ induce equi …

Let $R$ be a valuation ring, with fraction field $K$ and residue field $k$. Denote by $\Gamma=K^{\times}/R^{\times}$ the valuation group (assumed nontrivial). The valuation $v:K^{\ …

Is there an example of valued field in which any pseudo convergent set has a limit and such that this limit is unique?

Is it true that any two commuting endomorphisms of a complete discrete valued field extend to commuting automorphisms of the algebraic closure?

Let $R$ be a DVR with maximal ideal $xR$, and assume that $R$ is not complete in the $xR$-adic topology. Let $\hat{R}$ be the completion of $R$ in the $xR$-adic topology. Set $K=Q( …

What did Newton himself do, so that the "Newton polygon" method is named after him?