O teorema dos zeros de Hilbert e suas consequências

Abstract

In this work, we present concepts of commutative algebra and algebraic geometry, aiming to understand Hilbert’s Nullstellensatz. This is a famous result in the area of Algebra, particularly in Algebraic Geometry, which relates two concepts: radical ideals and affine varieties, establishing a correspondence between ideals and the set of zeros of polynomials (varieties). It is noteworthy that for any algebraically closed field K, there are bijections between algebraic varieties contained in the affine space An(K) and radical ideals contained in the polynomial ring K[x1, ..., xn]. The text is divided into three chapters: the first explores some fundamental concepts of commutative ring theory, the second presents basic notions of algebraic geometry, and the third addresses the theorem and its consequences.