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The phrase "Artin–Schreier theory" usually refers to a chapter in the theory of ordered fields (cf. Ordered field).
A field is formally real if the only solutions of
are
. 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
is real closed, then
is algebraically closed and, conversely, if
is algebraically closed and
, then
is real closed (the Artin–Schreier characterization of real-closed fields). A further Artin–Schreier theorem is that if
is the algebraic closure of
,
and
, then
is real closed and hence of characteristic zero and
.
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 be a field of real numbers, i.e. a subfield of the field of real numbers
, which has a unique ordering, and let
be a rational function (of several variables) with coefficients in
that is rationally definite in the sense that
for all
for which
is defined. Then
is a sum of squares of rational functions with coefficients in
.
References
[a1] | N. Jacobson, "Lectures in abstract algebra" , III: theory of fields and Galois theory , v. Nostrand (1964) pp. Chapt. VI |
[a2] | P. Ribenboim, "L'arithmétique des corps" , Hermann (1972) pp. Chapt. IX |
Artin-Schreier theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Artin-Schreier_theory&oldid=22031