Algebraic Construction of Quasi-split Algebraic Tori

TL;DR

This paper provides a constructive proof for a specific case of the no-name lemma, demonstrating that the invariant field of a quotient of a group algebra is purely transcendental, with explicit algebraically independent generators.

Contribution

It introduces a constructive method to identify generators of the invariant field for Galois groups acting on permutation lattices, extending the understanding of algebraic tori.

Findings

Constructed explicit algebraically independent generators for the invariant field.

Proved the invariant field is purely transcendental in the specific case.

Extended the no-name lemma to a new class of Galois actions.

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.