13A18-PlaceAsExtensionOfHomomorphism.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{PlaceAsExtensionOfHomomorphism}
\pmcreated{2013-03-22 14:57:21}
\pmmodified{2013-03-22 14:57:21}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{place as extension of homomorphism}
\pmrecord{9}{36651}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{13A18}
\pmclassification{msc}{12E99}
\pmclassification{msc}{13F30}
\pmsynonym{extension theorem}{PlaceAsExtensionOfHomomorphism}
\pmrelated{RamificationOfArchimedeanPlaces}

\endmetadata

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\newtheorem*{thmplain}{Theorem}
\begin{document}
\begin{thmplain}
\,If $f$ is a ring homomorphism from a subring $\mathfrak{o}$ of a field $k$ to an algebraically closed field $F$ such that \,$f(1) = 1$, \,then there exists a \PMlinkname{place}{PlaceOfField} 
              $$\varphi: \,k\to F\cup\{\infty\}$$
of the field $k$ such that
                  $$\varphi|_\mathfrak{o} = f.$$
\end{thmplain}

\textbf{Note.} \,That $F$ should be algebraically closed, does not \PMlinkescapetext{mean any restriction}, since every field is extendable to such one.

\begin{thebibliography}{8}
\bibitem{artin}Emil Artin: {\em \PMlinkescapetext{Theory of Algebraic Numbers}}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959).
\end{thebibliography}
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\end{document}