Between algebra and geometry: Solving systems of polynomial équations

Abstract

In the realm of factorial rings, we introduce the concept of the greatest common divisor (g.c.d.) for two elements, defined up to a unit, alongside the notion of prime elements. More broadly, Bézout's identity allows us to characterize pairs of prime elements within a principal ring, leading us to define these rings as Bézout rings.
Additionally, we delve into several geometric theorems that can be proven through algebraic methods, though the elegance of geometric approaches—especially in projective geometry—remains unparalleled. It is intriguing to note that many geometric challenges can be recast as polynomial equations, permitting us to frame them in terms of polynomial ideals. We present a variety of examples that highlight this connection, without attempting to construct an overarching theory.