The number of ramified primes in number fields of small degree

In this paper we investigate the distribution of the number of primes which ramify in number fields of degree d ≤ 5 d ≤ 5 . In analogy with the classical Erdős-Kac theorem, we prove for S d S_d -extensions that the number of such primes is normally distributed with mean and variance log ⁡ log ⁡ X log log X .