Artin-Schreier theory

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 $\sqrt{-1} \in F$, 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 , v. Nostrand (1964) pp. Chapt. VI
[a2]	P. Ribenboim, "L'arithmétique des corps" , Hermann (1972) pp. Chapt. IX