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

TL;DR

This paper offers an explicit method to construct the birational isomorphism in Weil's Galois descent theorem, enabling the definability of algebraic varieties over base fields in a computational manner.

Contribution

It provides a constructive approach to Weil's Galois descent theorem, which was previously non-constructive.

Findings

Explicit construction of the birational isomorphism R

Enhanced understanding of descent conditions in computational algebraic geometry

Potential applications in algorithmic algebraic geometry

Abstract

Let ${\mathcal L}/{\mathcal K}$ 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 $R.$ The aim of this paper is to give an explicit construction of $R$.