The integral closure of a primary ideal is not always primary

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=x6+y6+x4zt+z3 given by Briançon and Speder in 1975 is a counterexample to Krull's question.