The integral closure of a primary ideal is not always primary

Abstract

In 1936, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic $p=2$. We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal J of the polynomial $f=x^6+y^6+x^{4}zt+z^3$ given by Briançon and Speder (1975) is a counterexample to Krull's question.