Narrow Results

The main purpose of this work is to give a constructive proof for a particular case of the no-name lemma. Let $G$ be a finite group, $K$ a field that is equipped with a faithful $G$-action, and $L$ a sign permutation $G$-lattice (see the Introduction for the definition). Then $G$ acts naturally on the group algebra $K[L]$ of $L$ over $K$, and hence also on the quotient field $K(L)=Q(K[L])$. A well-known variant of the no-name lemma asserts that the invariant sub-field $K{(L)}^G$ is a purely transcendental extension of $K^G$. In other words, there exist $y_1,\dots ,y_n$ which are algebraically independent over $K^G$ such that $K{(L)}^G\cong K^G(y_1,\dots ,y_n)$. In this paper, we give an explicit construction of suitable elements $y_1,\dots ,y_n$.

In this paper, we give some topological properties and estimates of orbit of certain subsets of $K_v$-points of varieties under actions of algebraic tori. These results are concerned with an analogue of Bruhat-Tits’ question on the set of $v$-adic integral points of algebraic tori.