Fans, real valuations, and hereditarily-Pythagorean fields

Characterization of Fans in *-fields

In a formally real field, a fan is a certain special intersection of positive cones of orderings. Fans are very important to the study of quadratic forms over the field. The current status of the theory for commutative fields is well summarized in [9]. Our interest here will be in a *-field (D, *); that is, a skew field D with an involution *. The concept of *-ordering is due to Holland [8], th...

On Galois Groups over Pythagorean and Semi-real Closed Fields* by Ido Efrat

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 fie...

∗-Valuations and Hermitian Forms on Skew Fields

This paper is a survey of the literature on ways in which the concept of ordering can be extended to the setting of a division ring with involution and the main results for these extensions.

Orderings, Valuations and Hermitian Forms over ∗-fields

Forking and Dividing in Fields with Several Orderings and Valuations

variety). For each i, let φi(x) be a C-dense quantifier-free Li-formula with parameters from K. Then we can find a K-definable rational function f : C → P which is non-constant, and has the property that the divisor f−1(0) is a sum of distinct points in ⋂n i=1 φi(K), with no multipliticities. (In particular, the support of the divisor contains no points from C(K)\C(K) and no points from C \ C.)...