Artin-Schreier theory

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