On a generalized Artin-Schreier theorem for real-maximal fields (Q1320975)

A field is called real-maximal if it has no proper separable algebraic extension to which all its orderings extend. It is proved that a field \(K\) with \(0\leq e\leq 3\) orderings is real-maximal if and only if its absolute Galois group \(G(K)\) is a free pro-2 product of \(e\) copies of \(\mathbb{Z}/2\mathbb{Z}\). This generalizes results of \textit{E. Artin} and \textit{O. Schreier} \((e=1)\) [Abh. Math. Semin. Univ. Hamb. 5, 225-231 (1927)] and of \textit{S. Bredikhin}, \textit{Yu. Ershov} and \textit{V. E. Kal'nei} \((e=2)\) [Math. Notes 7, 319-325 (1970); translation from Mat. Zametki 7, 525-536 (1970; Zbl 0195.328)]. Counterexamples are given for any \(e\geq 4\). It is proved that if the orderings on a real-maximal field \(K\) are independent, then \(G(K)\) is a free pro-2 product of groups of order 2 indexed on the space of orderings of \(K\). It is shown that the only absolute Galois groups that force a field to be pseudo real closed are 1 and \(\mathbb{Z}/2\mathbb{Z}\).