On the Pythagorean hull of Q.
The purpose of this paper is to prove that the Pythagorean hull of Q is the splitting field in R of a special set of rational polynomials.
On the seperation of basic semialgebraic sets by polynomials.
On Trace Forms of Algebraic Number Fields.
On Witt rings of function fields of real analytic surfaces and curves.
Let V be a paracompact connected real analytic manifold of dimension 1 or 2, i.e. a smooth curve or surface. We consider it as a subset of some complex analytic manifold VC of the same dimension. Moreover by a prime divisor of V we shall mean the irreducible germ along V of a codimension one subvariety of VC which is an invariant of the complex conjugation. This notion is independent of the choice of the complexification VC. In the one-dimensional case prime divisors are just points, in the two-dimensional...
Orthogonality and complementation in the lattice of subspaces of a finite vector space
We investigate the lattice of subspaces of an -dimensional vector space over a finite field with a prime power together with the unary operation of orthogonality. It is well-known that this lattice is modular and that the orthogonality is an antitone involution. The lattice satisfies the chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when is orthomodular. For...
Pathological functions on Puiseux series ordered fields and others.
Positive polynomials and polynomial inequalities.
Préhistoire de la géométrie algébrique réelle : de Descartes à Tarski
Quadratic forms and radicals of fields
Quadratische Formen und die Artin-Schreiersche Theorie der formal reellen Körper
Quadratische Semi-Ordnungen und Quadratische Formen.
Quotients of index two and general quotients in a space of orderings
We investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field ℚ(x) is particularly interesting, since then the theory developed simplifies significantly. A part of the theory firstly developed for quotients of index 2 generalizes...
Rational points on algebraic varieties over a generalized real closed field: a model theoretic approach.
Real commutative algebra (I). Places.
Real commutative algebra (II). Plane curves.
Real commutative algebra. III. Dedekind-Weber-Riemann manifolds.
The space S of all non-trivial real places on a real function field K|k of trascendence degree one, endowed with a natural topology analogous to that of Dedekind and Weber's Riemann surface, is shown to be a one-dimensional k-analytic manifold, which is homeomorphic with every bounded non-singular real affine model of K|k. The ground field k is an arbitrary ordered, real-closed Cantor field (definition below). The function field K|k is thereby represented as a field of real mappings of S which might...
Rekursive Unentscheidbarkeit der Theorie der pythagoräischen Körper
Remarks on the Pythagoras and Hasse number of real fields.
Residual fields in valuation theory.
Residue class rings of real-analytic and entire functions
Let 𝓐(ℝ) and 𝓔(ℝ) denote respectively the ring of analytic and real entire functions in one variable. It is shown that if 𝔪 is a maximal ideal of 𝓐(ℝ), then 𝓐(ℝ)/𝔪 is isomorphic either to the reals or a real closed field that is an η₁-set, while if 𝔪 is a maximal ideal of 𝓔(ℝ), then 𝓔(ℝ)/𝔪 is isomorphic to one of the latter two fields or to the field of complex numbers. Moreover, we study the residue class rings of prime ideals of these rings and their Krull dimensions. Use is made of...