An elementary proof for the Krull dimension of a polynomial ring

TL;DR

This paper provides an elementary proof that the Krull dimension of a polynomial ring over an infinite field is equal to the number of variables, using only basic algebraic concepts.

Contribution

It offers a simplified, accessible proof of a fundamental algebraic fact, avoiding advanced techniques.

Findings

Krull dimension of polynomial rings over infinite fields is n

Elementary proof using basic algebra suffices

Accessible approach for educational purposes

Abstract

This is an expository paper in which it is proved that, for every infinite field ${\mathbf{F}}$, the polynomial ring ${\mathbf{F}}[t_1,\ldots, t_n]$ has Krull dimension $n$. The proof uses only "high school algebra" and the rudiments of undergraduate "abstract algebra."