Algebraic Construction of Quasi-split Algebraic Tori

Abstract

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, $K$ be a field, $L$ be a permutation $G$-lattice and $K[L]$ be the group algebra of $L$ over $K$. The no-name lemma asserts that the invariant field of the quotient field of $K[L]$, $K(L)^G$ is a purely transcendental extension of $K^G$. In other words, there exist $y_1, \ldots , y_n$ which are algebraically independent over $K^G$ such that $K(L)^G \cong K^G(y_1, \ldots , y_n)$. We define elements $\lbrace y_1, \ldots, y_n \rbrace \subset K[L]^G$ with the desired properties, in the case when $G$ is the Galois group of a finite extension $\mathrm{Gal}(K/F)$, and $L$ is a sign permutation $G$-lattice.