A generalization of Marshall’s equivalence relation

Abstract

Abstract. For p prime and for a field F containing a root of unity of order p, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of F × of index p. The equivalence classes then correspond to free pro-p factors of the maximal pro-p Galois group of F . We generalize to this setting results of Jacob on the maximal pro-2 Galois group of a Pythagorean field.