Let be a prime number, the field of -adic numbers and the completion of the algebraic closure of . In this paper we obtain a representation theorem for rigid analytic functions on which are equivariant with respect to the Galois group , where is a lipschitzian element of and denotes the -neighborhood of the -orbit of .
This paper is devoted to some variants of the Hilbert specialization property. For example, the RG-hilbertian property (for a field K), which arose in connection with the Inverse Galois Problem, requires that the specialization property holds solely for extensions of K(T) that are Galois and regular over K. We show that fields inductively obtained from a real hilbertian field by adjoining real pth roots (p odd prime) are RG-hilbertian; some of these fields are not hilbertian. There are other variants...
Let be a Galois extension with Galois group . We study the set of -linear combinations of characters in the Burnside ring which give rise to -linear combinations of trace forms of subextensions of which are trivial in the Witt ring W of . In particular, we prove that the torsion subgroup of coincides with the kernel of the total signature homomorphism.