Abstract
Let k be a real field, Ω its algebraic closure and G=Gal(Ω;k) its absolute Galois group. G can be very large and so one looks for simple identifiable types of small closed subgroups H of G containing involutions so that the corresponding fixed fields of H are all real.
From Artin-Schreier Theory, we know that the simplest such type are groups of order 2. Looking for larger groups, we construct some interesting examples by “digging holes” into real closed fields.
Let R be a real closed field and α some fixed element of R. Let k be a subfield of R maximal with respect to the exclusion of α. In this paper we give a complete description of k and Gal(Ω;k). We also prove that our description of Gal(Ω;k) characterizes Gal(Ω;k) as the total Galois group of a field maximal excluding some element in some of its real closures.
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Engler, A.J., Viswanathan, T.M. Formally real fields with a simple description of the absolute Galois group. Manuscripta Math 56, 71–87 (1986). https://doi.org/10.1007/BF01171034
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DOI: https://doi.org/10.1007/BF01171034