On Galois Groups over Pythagorean and Semi-real Closed Fields* by Ido Efrat
نویسنده
چکیده
We call a field K semi-real closed if it is algebraically maximal with respect to a semi-ordering. It is proved that (as in the case of real closed fields) this is a Galois-theoretic property. We give a recursive description of all absolute Galois groups of semi-real closed fields of finite rank. I n t r o d u c t i o n By a well-known theorem of Artin and Schreier [AS], being a real closed field is a Galois-theoretic property. More specifically, a field K is real closed if and only if its absolute Galois group G(K) is of order two. This enables one to reflect many arithmetical properties of orderings on K as group-theoretic properties of G(K). However, in studying the structure of formally real fields, the collection of all orderings is in many respects too small. For many uses one needs to consider the * Research carried out at the Institute for Advanced Studies of the Hebrew University of Jerusalem. Received March 2, 1992 and in revised form August 19, 1993
منابع مشابه
On Fields with Finite Brauer Groups
Let K be a field of characteristic 6= 2, let Br(K)2 be the 2primary part of its Brauer group, and let GK(2) = Gal(K(2)/K) be the maximal pro-2 Galois group of K. We show that Br(k)2 is a finite elementary abelian 2-group (Z/2Z), r ∈ N, if and only if GK(2) is a free pro-2 product of a closed subgroup H which is generated by involutions and of a free pro-2 group. Thus, the fixed field of H in K(...
متن کاملRecovering higher global and local fields from Galois groups — an algebraic approach
We consider the following general problem: let F be a known field with absolute Galois group GF . Let K be a field with GK ≃ GF . What can be deduced about the arithmetic structure of K? As a prototype of this kind of questions we recall the celebrated Artin–Schreier theorem: GK ≃ GR if and only if K is real closed. Likewise, the fields K with GK ≃ GE for some finite extension E of Qp are the p...
متن کاملProjective extensions of fields
A field K admits proper projective extensions, i.e. Galois extensions where the Galois group is a nontrivial projective group, unless K is separably closed or K is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maxi...
متن کاملIntersections of Real Closed Fields
1. In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We s...
متن کامل