Faster computation of the first factor of the class number of 𝑄(𝜁_{𝑝})

Abstract

We describe two fast methods for computing the first factor of the class number of the cyclotomic field Q ( ζ p ) \mathbb {Q}({\zeta _p}) in O ( p 2 log 5 p ) \mathcal {O}({p^2}{\log ^5}p) and O ( p 2 log ⁡ p ) \mathcal {O}({p^2}\log p) steps of elementary arithmetic operations on the numbers of size p, respectively. The first is deterministic, while the second holds under the GRH. This is an improvement over the previous method of Lehmer and Masley, which has complexity O ( p 3.81 ) \mathcal {O}({p^{3.81}}) .