On the field of definition of vector bundles on real varieties (Q685201)

In principle this paper is a continuation (with other tools) of a joint paper of the author with \textit{A. Tognoli} [Geom. Dedicata 42, No. 2, 155- 161 (1992; Zbl 0762.14029)]. We consider the possibility of defining over small fields the generators for the \(K\)-theory of strongly algebraic vector bundles on a real smooth variety. Furthermore we discuss how to construct in an explicit way algebraic models (defined over small fields and with other good arithmetic properties) of two-dimensional disconnected differential manifolds (and related singular spaces). The main result of this paper is: Theorem 1.3. Let \(Y\) be a smooth algebraic model of the compact differential manifold \(M\), with \(Y\) defined over a field \(K \subset \mathbb{R}\). Fix a prime \(p\) and let \(K\{p\}\) be the algebraic extension field of \(K\) contained in \(\mathbb{R}\) and generated by all finite extensions of \(K\) with order a \(p\)-power and contained in \(\mathbb{R}\). Then there is a finite group \(G\) without normal subgroups \(\neq \{\text{Id}\}\) with order prime to \(p\), and a Galois extension \(L\) of \(K\{p\}\) with Galois group \(G\) such that the part, \(T(Y)\), of \(K(X)_{\text{alg}} \otimes \mathbb{Z}_ p\) coming from \(K(Y)\) comes from \(K(Y;!K \{p\})^ G\).