On the computation of the class number of an algebraic number field

Abstract

It is shown how the analytic class number formula can be used to produce an algorithm which efficiently computes the class number h of an algebraic number field F. The method assumes the truth of the Generalized Riemann Hypothesis in order to estimate the residue of the Dedekind zeta function of F at s = 1 s = 1 sufficiently well that h can be determined unambiguously. Given the regulator R of F and a known divisor h ∗ {h^ \ast } of h, it is shown that this technique will produce the value of h in O ( | d F | 1 + ε / ( h ∗ R ) 2 ) O(|{d_F}{|^{1 + \varepsilon }}/{({h^ \ast }R)^2}) elementary operations, where d F {d_F} is the discriminant of F. Thus, if h > | d F | 1 / 8 h > |{d_F}{|^{1/8}} , then the complexity of computing h (with h ∗ = 1 {h^ \ast } = 1 ) is O ( | d F | 1 / 4 + ε ) O(|{d_F}{|^{1/4 + \varepsilon }}) .