A General Version of the Nullstellensatz for Arbitrary Fields

TL;DR

This paper extends the Nullstellensatz to arbitrary fields, establishing a condition based on local roots that generalizes the classical algebraically closed field case.

Contribution

It provides a generalized Nullstellensatz applicable to any field, linking local root existence to ideal membership.

Findings

Generalized Nullstellensatz for arbitrary fields

Condition based on local multi-valued roots

Recovers standard Nullstellensatz over algebraically closed fields

Abstract

We prove a general version of Bezout's form of the Nullstellensatz for arbitrary fields. The corresponding sufficient and necessary condition only involves the local existence of multi-valued roots for each of the polynomials belonging to the ideal in consideration. Finally, this version implies the standard Nullstellensatz when the coefficient field is algebraically closed.