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\begin{document}

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\section*{real closed field}

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{infinites_and_infinitesimals}{Infinites and infinitesimals}\dotfill \pageref*{infinites_and_infinitesimals} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

An [[ordered field]] $F$ is \textbf{real closed} if it satisfies the following two properties:

\begin{itemize}%
\item Any positive element $x \geq 0$ in $F$ has a [[square root]] in $F$;


\item Any odd-degree [[polynomial]] with coefficients in $F$ has a root in $F$.



\end{itemize}
Notice that the order on a real closed field is definable from the algebraic structure: $x \leq y$ if and only if $\exists_z x + z^2 = y$. (In particular, there is a \emph{unique} ordering on a real closed field, defined by taking the positive elements to be precisely the nonzero squares.) In fact, the [[category]] of real closed fields and order-preserving field homomorphisms is a [[full subcategory]] of the category of [[fields]] and field homomorphisms.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Real closed fields can be equivalently characterized by any of the following properties:

\begin{enumerate}%
\item $F$ is not [[algebraically closed field|algebraically closed]], but some finite [[field extension|extension]] is. This extension is necessarily $F[\sqrt{-1}]$. See also [[fundamental theorem of algebra]].


\item As a field, $F$ is [[elementary equivalence|elementarily equivalent]] to the field of real numbers.


\item The [[intermediate value theorem]] holds for all polynomials with coefficients in $F$.


\item $F$ is an ordered field that has no ordered algebraic extension.



\end{enumerate}
In fact, there is a [[completion]] of any [[ordered field]] to a real closed field, in the following sense:

\begin{utheorem}
The full inclusion of the category of real closed fields and field homomorphisms to the category of ordered fields and ordered field homomorphisms has a left adjoint.

\end{utheorem}
\begin{proof}
We give a brief sketch of proof, referring to Lang's \emph{Algebra} ($3^{rd}$ edition), section IX.2, for more details.

First, for each ordered field $F$, there is a real closed algebraic extension $F \to R$ that is order-preserving (theorem 2.11). This is called a \textbf{real closure} of the ordered field $F$.

Second, any two real closures of $F$ are uniquely isomorphic (theorem 2.9); in fact, the proof shows there is at most one order-preserving homomorphism over $F$ between any two real closures. Therefore we may speak of \emph{the} real closure of $F$, which we denote as $\widebar{F}$.

Finally, let $F \to R$ be any order-preserving field homomorphism to a real closed field $R$. We must show that $F \to R$ extends uniquely to a homomorphism $i: \widebar{F} \to R$. Any such homomorphism $i$ must factor through the subfield $R' \hookrightarrow R$ consisting of elements $\alpha \in R$ that are algebraic over $F$, since $\widebar{F}$ is algebraic over $F$. But this subfield is also real closed. Therefore, by the preceding paragraph, there is at most one homomorphism $\widebar{F} \to R'$ extending $F \to R'$, and the proof is complete.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{enumerate}%
\item The [[real number]]s form a real closed field.


\item Real [[algebraic number]]s form a real closed field, which is the real closure of the ordered field of [[rational numbers]].


\item A field of nonstandard real numbers (as in Robinson [[nonstandard analysis]]) is real closed.


\item [[surreal number|Surreal numbers]] form a (large) real closed field.


\item If $F$ is real closed, then the field of [[Puiseux series]] over $F$ is also real closed.


\item More generally, given a real closed field $F$, the field of [[Hahn series]] over $F$ with [[valuation ring|value group]] $G$ (a linearly ordered group) is real closed provided that $G$ is [[divisible group|divisible]].


\item Any [[o-minimal structure|o-minimal]] ordered ring structure $R$ is a real closed field.


\item Given an o-minimal ordered ring $R$, the field of [[germ]]s at infinity of definable functions $R \to R$ in any o-minimal expansion of $(R, 0, 1, +, -, \cdot, \lt)$ is real closed. (By ``germ at infinity'', we mean an equivalence class of functions for which $f \equiv g$ if and only if $f(x) = g(x)$ for all sufficiently large $x$.)



\end{enumerate}
\hypertarget{infinites_and_infinitesimals}{}\subsection*{{Infinites and infinitesimals}}\label{infinites_and_infinitesimals}

Each real closed field $R$ contains a [[valuation ring|valuation]] subring $B \hookrightarrow R$ consisting of the ``bounded'' or archimedean elements, i.e., elements $x \in R$ such that $-n \leq x \leq n$ for some integer multiple $n$ of the identity. An element in the complement of $B$ is an \textbf{infinite} element of $R$, and the reciprocal of an infinite element is an \textbf{infinitesimal} element. The field of fractions of $B$ is clearly $R$.

We remark that any real closed field contains a copy of the field of real [[algebraic numbers]], which as before we denote by $\widebar{\mathbb{Q}}$ (not to be confused with the algebraic closure of $\mathbb{Q}$). Each of the elements of $\widebar{\mathbb{Q}}$ is archimedean.

Let $B^\ast$ be the group of units of $B$. The quotient $R^\ast/B^\ast$ is the \textbf{value group} of $R$. It can be viewed as the ``group of orders of infinities and infinitesimals'' of $R$. If $R$ is real closed, then the value group is a linearly ordered [[divisible group]] (divisible because we can take $n^{th}$ roots of positive elements in $R$). The structure of the value group as ordered group is an important invariant of the real closed field.

In the other direction, to each ordered divisible abelian group $G$, there exists a real closed field having $G$ as its value group. For example, one may form the [[Hahn series]] over $\widebar{\mathbb{Q}}$ with value group $G$.

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

\begin{itemize}%
\item \href{Euclidean+geometry#TarskiAxioms}{Euclidean geometry -- Tarski's axioms}

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Serge Lang]], \emph{Algebra} (3rd edition), Addison-Wesley, 1993.


\item David Marker, \emph{Notes on Real Algebra} \href{http://www.math.uic.edu/~marker/orsay/real_algebra.pdf}{(link)}



\end{itemize}
[[!redirects real closed field]] [[!redirects real closed fields]] [[!redirects real-closed field]] [[!redirects real-closed fields]]



\end{document}