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Difference between revisions of "Artin-Schreier theory"

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The phrase  "Artin–Schreier theory"  usually refers to a chapter in the theory of ordered fields (cf. [[Ordered field|Ordered field]]).
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The phrase  "Artin–Schreier theory"  usually refers to a chapter in the theory of [[ordered field]]s.
  
A field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107501.png" /> is formally real if the only solutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107502.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107503.png" />. Any such field can be ordered and, conversely, any ordered field is formally real. A real-closed field is a formally real field that is maximal under algebraic extensions. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107504.png" /> is real closed, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107505.png" /> is algebraically closed and, conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107506.png" /> is algebraically closed and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107507.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107508.png" /> is real closed (the Artin–Schreier characterization of real-closed fields). A further Artin–Schreier theorem is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a1107509.png" /> is the algebraic closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075013.png" /> is real closed and hence of characteristic zero and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075014.png" />.
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A [[formally real field]] $F$ has the property that the only solutions of $x_1^2 + \cdots + x_n^2 = 0$ are $x_1 = \cdots = x_n = 0$. Any such field can be ordered and, conversely, any ordered field is formally real. A [[real-closed field]] is a formally real field that is maximal under algebraic extensions. If $F$ is real closed, then $F(\sqrt{-1})$ is algebraically closed and, conversely, if $F(\sqrt{-1})$ is algebraically closed and $\sqrt{-1} \not\in F$, then $F$ is real closed (the Artin–Schreier characterization of real-closed fields). A further Artin–Schreier theorem is that if $\bar F$ is the algebraic closure of $F$, $\bar F \neq F$ and $[\bar F:F] < \infty$, then $F$ is real closed and hence of characteristic zero and $\bar F = F(\sqrt{-1})$.
  
The theory of formally real fields led E. Artin to the solution of the Hilbert problem on the resolution of definite rational functions as sums of squares (the Artin theorem): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075015.png" /> be a field of real numbers, i.e. a subfield of the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075016.png" />, which has a unique ordering, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075017.png" /> be a rational function (of several variables) with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075018.png" /> that is rationally definite in the sense that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075020.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075021.png" /> is defined. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075022.png" /> is a sum of squares of rational functions with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110750/a11075023.png" />.
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The theory of formally real fields led E. Artin to the solution of the Hilbert problem on the resolution of definite rational functions as sums of squares (the Artin theorem): Let $F$ be a field of real numbers, i.e. a subfield of the field of real numbers $\mathbf{R}$, which has a unique ordering, and let $Q$ be a rational function (of several variables) with coefficients in $F$ that is rationally definite in the sense that $Q(x_1,\ldots,x_n) \ge 0$ for all $x_1,\ldots,x_n$ for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with coefficients in $F$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''III: theory of fields and Galois theory''' , v. Nostrand  (1964)  pp. Chapt. VI</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ribenboim,  "L'arithmétique des corps" , Hermann  (1972)  pp. Chapt. IX</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Jacobson,  "Lectures in abstract algebra" , '''III: theory of fields and Galois theory''' , van Nostrand  (1964)  pp. Chapt. VI {{ZBL|0124.27002}}</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ribenboim,  "L'arithmétique des corps" , Hermann  (1972)  pp. Chapt. IX {{ZBL|0253.12101}}</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 20:06, 21 September 2017

The phrase "Artin–Schreier theory" usually refers to a chapter in the theory of ordered fields.

A formally real field $F$ has the property that the only solutions of $x_1^2 + \cdots + x_n^2 = 0$ are $x_1 = \cdots = x_n = 0$. Any such field can be ordered and, conversely, any ordered field is formally real. A real-closed field is a formally real field that is maximal under algebraic extensions. If $F$ is real closed, then $F(\sqrt{-1})$ is algebraically closed and, conversely, if $F(\sqrt{-1})$ is algebraically closed and $\sqrt{-1} \not\in F$, then $F$ is real closed (the Artin–Schreier characterization of real-closed fields). A further Artin–Schreier theorem is that if $\bar F$ is the algebraic closure of $F$, $\bar F \neq F$ and $[\bar F:F] < \infty$, then $F$ is real closed and hence of characteristic zero and $\bar F = F(\sqrt{-1})$.

The theory of formally real fields led E. Artin to the solution of the Hilbert problem on the resolution of definite rational functions as sums of squares (the Artin theorem): Let $F$ be a field of real numbers, i.e. a subfield of the field of real numbers $\mathbf{R}$, which has a unique ordering, and let $Q$ be a rational function (of several variables) with coefficients in $F$ that is rationally definite in the sense that $Q(x_1,\ldots,x_n) \ge 0$ for all $x_1,\ldots,x_n$ for which $Q(x_1,\ldots,x_n)$ is defined. Then $Q$ is a sum of squares of rational functions with coefficients in $F$.

References

[a1] N. Jacobson, "Lectures in abstract algebra" , III: theory of fields and Galois theory , van Nostrand (1964) pp. Chapt. VI Zbl 0124.27002
[a2] P. Ribenboim, "L'arithmétique des corps" , Hermann (1972) pp. Chapt. IX Zbl 0253.12101
How to Cite This Entry:
Artin-Schreier theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_theory&oldid=22031
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article