Fields: Algebraically closed and others
This expository paper contains a detailed study of the fields \([[F^{S,\leq}]]\) of generalized power series with the exponents in a totally ordered torsion-free group. It includes a self-contained introduction to the subject and the selection of various results about general power series. The main topics considered by the author refer to algebraic extensions, algebraically closed fields, sums of squares and real fields, Pythagorean fields and Euclidean fields. The paper contains many interesting new results and significant improvements of others. We mention, for example, the description of an algebraic closure of the generalized power series with the exponents in a torsion-free group in the case of characteristic zero, respectively in a totally ordered \(p\)- divisible group for fields of characteristic \(p\). There are discussed other results about the algebraic closure, including those of \textit{F. J. Rayner} [J. Lond. Math. Soc., II. Ser. 8, 504-506 (1974; Zbl 0306.12103)]. The paper includes a characterization of the Pythagorean generalized power series fields with coefficients in a field of characteristic different from 2 and of the Pythagorean closure of such fields. Results of \textit{A. Benhissi} [C. R. Math. Acad. Sci., Soc. R. Can. 12, 193-198 (1990; Zbl 0731.12007)] are extended and a description of the Euclidean closure of the field of general power series over a real Euclidean field is obtained. The paper contains many comments about the various results obtained by other authors. There is also included a selected bibliography.
Pythagorean generalized power series fields
scientific article; zbMATH DE number 91455
scientific article