Quintic Fields

We had to invest the best known reduction and enumeration algorithms, many new ideas and our fastest PC-s to be able to solve index form equations in quintic fields. In the interesting case, for totally real quintic fields (with Galois group A 5 or S 5 ) this computation takes several hours, contrary to the cubic and quartic cases, where to solve the index form equation was a matter of seconds or at most some minutes. The general method is described in Section 7.1. Having read the relatively complicated formulas of this procedure, in Section 7.2 the reader is rewarded with an interesting family of totally real cyclic quintic fields introduced by E.Lehmer.