Let \(F\) be a formally real field and \(P\) the positive cone of an ordering on \(F\). It was shown more than 60 years ago by Artin and Schreier, that the maximal algebraic extension of \(F\), to which the ordering \(P\) extends, the real closure of \(F\) with respect to \(P\), is unique in an order preserving way.
E. Becker in “Hereditarily-Pythagorean fields and orderings of higher level” [IMPA Monograf. Mat. No. 29 (1978; Zbl 0509.12020)] generalized the notion of orderings as follows: If \(F\) is a field with multiplicative group \(\dot F\), then \(P=\dot P\cup \{0\}\), is an ordering of higher level \(m\), if \(\dot F^{2^m}\subset P\), \(\dot P\) is subgroup of \(\dot F\) with \(\dot F/\dot P\) cyclic of even order, and \((\dot P+\dot P)\subset \dot P\). Becker (loc. cit.) showed that there again are maximal algebraic extension fields of \(F\) to which \(P\) extends, so-called real closures of higher level, which, however, are highly non-unique. This was remedied by J. Harman in [Contemp. Math. 8, 141–174 (1982; Zbl 0509.12021)], who showed that there is a bijection between real closed fields of higher order and certain sequences of orderings of higher level called “chains”.
In the paper under review the author studies these chains and shows that this study can be reduced to tractable problems in abelian group theory and in the theory of real closures of level 1, i.e., the real closed fields introduced by Artin and Schreier. Moreover, the author’s methods enable him to study chains under field extensions, subfield formation, and places. There is a paper by N. Schwartz dealing with related questions [J. Algebra 110, 74–107 (1987; Zbl 0632.12022)] which, it should be noted, was submitted over three years after the paper under review was.