Periodicity in Cubic Number Field

Jacob Jacobi in 1868 published his studies in Number Theory. Professor Fritz Schweiger writes:

"Maybe he was warned off by the long calculations for (1; 5^¹/³; 25^¹/³) and the obviously never ending calculations on (1; 4^¹/³; 16^¹/³). In fact, it is still not known if periodicity occurs at all!"

I am curious to understand the problems Jacobi run into.