Weil's Galois Descent Theorem from a computational point of view

Let $\mathcal{L}\mathrm{/}\mathcal{K}$ be a finite Galois extension and let $X$ be an affine algebraic variety defined over $\mathcal{L}$. Weil’s Galois descent theorem provides necessary and sufficient conditions for $X$ to be definable over $\mathcal{K}$, that is, for the existence of an algebraic variety $Y$ defined over $\mathcal{K}$ together with a birational isomorphism $R:X\to Y$ defined over $\mathcal{L}$. Weil’s proof does not provide a method to construct the birational isomorphism $R\mathrm{.}$ The aim of this paper is to give an explicit construction of $R$.