henselian field HenselianField 2003-02-22 04:33:08 2007-06-02 12:28:56 Definition valuation ring residue field residue class field Hensel property henselian henselisation hensel valuation non archimedean % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{amssymb} \usepackage{eufrak} \usepackage[dvips]{epsfig,graphics} \usepackage{graphicx,psfrag} % Theorems \newtheorem{df}{Definition}[section] \newtheorem{tm}[df]{Theorem} \newtheorem{lm}[df]{Lemma} \newtheorem{crl}[df]{Corollary} \newtheorem{pp}[df]{Proposition} % Spelling issues \def\centre{\center} % Textrm commands % \def\bugger{{\rm bugger}} \def\Int{{\rm Int}_{k,C}} \def\Intn{{\rm Int}_{k,C}^{n}} \def\acl{{\rm acl}} \def\ch{{\rm char}} \def\dcl{{\rm dcl}} \def\dom{{\rm dom}} \def\elt{{\rm elt}} \def\eut{{\rm eut}} \def\fiber{{\rm fib}} \def\germ{{\rm germ}} \def\graph{{\rm graph}} \def\gr{{\rm grd}} \def\ilt{{\rm ilt}} \def\lit{{\rm ilt}} \def\iut{{\rm iut}} \def\lex{{\rm lex}} \def\mo{{\rm m.o.}} \def\rad{{\rm rad}} \def\red{{\rm red}} \def\rex{{\rm rex}} \def\rcl{{\rm rcl}} \def\tp{{\rm tp}} \def\Aff{{\rm Aff}} \def\Hom{{\rm Hom}} \def\Res{{\rm Res}} \def\Gl{{\rm Gl}} % Math frac commands \newcommand{\ma}{\mathfrak{A}} \newcommand{\mb}{\mathfrak{B}} \newcommand{\mc}{\mathfrak{C}} \newcommand{\md}{\mathfrak{D}} % Other commands \newcommand{\acf}{ACF_{val}} \newcommand{\bm}{\begin{displaymath}} \newcommand{\cl}{\bf{d}} \newcommand{\fp}{f^{\prime}} \newcommand{\gda}{G_{D}^{A}} \newcommand{\gind}{\downarrow^{g}} \newcommand{\op}{\bf{o}} \newcommand{\half}{{\tiny \begin{array}{l}1 \\ \overline{2}\end{array}}} \def\isom{\simeq} \newcommand{\mgb}{\mathbf{M}} \newcommand{\nin}{\not\in} \def\nom{\vartriangleleft} \newcommand{\ra}{\rightarrow} \newcommand{\rcf}{RCVF_{G}} \newcommand{\real}{\textrm{real}} \newcommand{\rk}{{\bf Remark:}} \newcommand{\sequ}{\left< x_{n}:x<\omega \right>} \newcommand{\sq}{ $\square$} \newcommand{\xb}{\overline{x}} \newcommand{\Aut}{\textrm{Aut}} \newcommand{\PR}{^{\prime}} \newcommand{\RS}{\mathbf{R}^{\star}} \newcommand{\LG}{L_{4}} \newcommand{\RR}{\mathbf{R}} \newcommand{\RA}{\rightarrow} \newcommand{\cla}[1]{\lceil #1 \rceil} %% \DeclareMathOperator{\res}{res} \newcommand{\val}{|\!\cdot\!|} Let $\val$ be a non-archimedean valuation on a field $K$. Let $V=\{x:|x|\le 1\}$. Since $\val$ is ultrametric, $V$ is closed under addition and in fact an additive group. The other valuation axioms ensure that $V$ is a ring. We call $V$ the \emph{valuation ring} of $K$ with respect to the valuation $\val$. Note that the field of fractions of $V$ is $K$. The set $\mu=\{x:|x|<1\}$ is a maximal ideal of $V$. The factor $R:=V/\mu$ is called the \emph{residue field} or the \emph{residue class field}. The map $\res:V \to V/\mu$ given by $x \mapsto x+\mu$ is called the \emph{residue map}. We extend the definition of the residue map to sequences of elements from $V$, and hence to $V[X]$ so that if $f(X) \in V[X]$ is given by $\sum_{i \leq n} a_{i}X^{i}$ then $\res(f) \in R[X]$ is given by $\sum_{i \leq n} \res(a{i})X^{i}$. \bigskip \par\noindent{\bf Hensel property:} Let $f(x) \in V[x]$. Suppose $\res(f)(x)$ has a simple root $e \in k$. Then $f(x)$ has a root $e\PR \in V$ and $\res(e\PR)=e$. \medskip Any valued field satisfying the Hensel property is called \emph{henselian}. The completion of a non-archimedean valued field $K$ with respect to the valuation (cf. constructing the reals from the rationals as the completion with respect to the standard metric) is a henselian field. Every non-archimedean valued field $K$ has a unique (up to isomorphism) smallest henselian field $K^h$ containing it. We call $K^h$ the \emph{henselisation} of $K$.