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\begin{document}

%-------------------------------------------------------------------


\section*{place}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \textbf{place} of a [[commutative unital ring]] has different meanings in the literature:

\begin{itemize}%
\item The most common one is an equivalence class of ([[archimedean field|archimedean]] or non-archimedean) [[absolute value]]s. (An \emph{absolute value} is a non-trivial multiplicative [[seminorm]].) This convention is often used in [[number theory]], where it applies to [[number field|number]] or [[function fields]].


\item It can mean an equivalence class of (possibly higher-rank) [[valuations]]. This convention is sometimes used in non-archimedean [[analytic geometry]], after Zariski's work (Zariski--Riemann space).


\item It can mean an equivalence class of morphism to [[fields]]. This convention is used in [[scheme]] theory, where a place is exactly the same thing as a [[prime ideal]] (easy proof).



\end{itemize}
Other notions of places can be imagined, combining the three above classical examples.

The notion of place in number theory is interesting because it is at the heart of the geometric study of [[zeta functions]]. An improvement on this notion is given by the setting of [[global analytic geometry]], which includes trivial seminorms that allow a natural geometric definition of [[adele]]s and [[idele]]s.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[place at infinity]]

\end{itemize}
[[!redirects place]] [[!redirects places]]



\end{document}