One of the fundamental questions in current field theory, related to Grothendieck’s conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the fields themselves. In this paper we initiate the classification of additive properties of multiplicative subgroups of fields containing all squares, using pro...

The theory of special groups is an axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia (see [4]). The class of special groups, together with its morphisms, forms a category. As for other such axiomatisations, the main examples of special groups are provided by fields, in this case by applying the special group functor, which associates to each field F ...

In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic forms when its adjoint involution decomposes as a tensor product of involutions on central simple algebras. We give a positive answer for quadratic forms defined over local or global fields and produce counterexamples over fields of rational fractions in two variable...

We present an approach for building highly-dependable systems that derive their assurance from a formally-verified operatingsystem which guarantees isolation between subsystems. We leverage those guarantees to enforce security through non-bypassable architectural constraints, and through generation of code and proofs from the architecture. We show that this approach can produce a system that is...

In order to demonstrably satisfy hard real-time deadlines, a system must be predictable, and in particular the kernel must be predictable. In this paper we present and analyse a predictable kernel related to AORTA, a formal design language for hard real-time systems. The features of the kernel allow AORTA designs to be veriiably and semi-automatically implemented, and enable veriied guarantees ...

Two important results in quadratic form theory, Pfister’s local-global principle and the isotropy theorem (see [5]), can be stated more generally for spaces of orderings (an abstract version of real spectras of formally real fields), for which they are expressed as local-global principles: A property of quadratic forms (expressed as a so-called positive-primitive formula) holds if and only if i...

The Bröcker–Prestel local-global principle characterizesweak isotropy of quadratic forms over a formally real field in terms of weak isotropy over the henselizations and isotropy over the real closures of that field. A hermitian analogue of this principle is presented for algebras of index at most two. An improved result is also presented for algebras with a decomposable involution, algebras of...

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...