Given fields k ? L , our results concern one parameter L-parametric polynomials over and their relation to generic polynomials. The former are P ( T Y ) ? [ ] of group G which parametrize all Galois extensions via specialization in the latter those -parametric for every field ? . We show, example, that being with taken be single C V U is fact sufficient a polynomial generic. As corollary, we ob...

Let K be a number field, let OK be the ring of integers, let K be an algebraic closure of K and let OK be the ring of integers of K. Let M 0 K be the set of finite places and let M∞ K be the set of infinite places. Let Kv be the completion of K at v and let Ov be the ring of integers of Kv. Let ℘v, kv, qv be the maximal ideal of Ov, the residue field Ov/℘v and the size of the residue field |kv|...

Let G be an algebraic group defined over a number field k. By choosing a lifting of G to a group scheme over 6' s c k, the ring of S-integers for some finite set of places S of k, we may define G(C,~), where (5~, c k~ is the ring of integers in the vadic completion of k for all non-archimedean places vr In this way, we can define the adelic points G(Ak). Since different choices of lifting will ...