Algebraic Construction of Quasi-split Algebraic Tori

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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 Kᵍ. In other words, there exist y₁,, yₙ which are algebraically independent over Kᵍ such that K(L)ᵍ Kᵍ(y₁,, yₙ). We define elements y₁,, yₙ ⊂ K[L]ᵍ with the desired properties, in the case when G is the Galois group of a finite extension Gal(K/F), and L is a sign permutation G-lattice.