So, there's a passing comment in the very first chapter of Atiyah-Macdonald saying:
If $K[x_1, \dots, x_n]$ is a polynomial ring, where $K$ is a field, then the ideal consisting of all polynomials with no constant term can not be generated by less than $n$ generators as an ideal.
Is this statement trivial and can it be proven in the framework of basic algebra? Or does it require some advanced machinery like dimension theory?