Explicit bounds for generators of the class group

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group Cl_K of a number field K may be generated using prime ideals whose norm is bounded by 12 log^2 Delta_K, and by (4+o(1))log^2 Delta_K asymptotically, where Delta_K is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates Cl_K and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that Cl_K is generated by prime ideals whose norm is bounded by the minimum of 4.01 log^2 Delta_K, 4(1+(2 pi e^gamma)^(-n_K))^2 log^2 Delta_K and 4(log Delta_K + log log Delta_K - (gamma+log 2\pi)n_K+1+(n_K+1)(log(7 log Delta_K)/log Delta_K)^2. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size of the order of (\log Delta_K/log log Delta_K)^2. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log^2 Delta_K except for 7 out of the 31292 fields we tested.