On the class numbers of real cyclotomic fields of conductor $pq$

Eleni Agathocleous¹

¹ Ministry of Education Nicosia, Cyprus

Acta Arithmetica, Tome 165 (2014) no. 3, pp. 257-277.

Résumé

The class numbers $h^{+}$ of the real cyclotomic fields are very hard to compute. Methods based on discriminant bounds become useless as the conductor of the field grows, and methods employing Leopoldt's decomposition of the class number become hard to use when the field extension is not cyclic of prime power. This is why other methods have been developed, which approach the problem from different angles. In this paper we extend one of these methods that was designed for real cyclotomic fields of prime conductor, and we make it applicable to real cyclotomic fields of conductor equal to the product of two distinct odd primes. The main advantage of this method is that it does not exclude the primes dividing the order of the Galois group, in contrast to other methods. We applied our algorithm to real cyclotomic fields of conductor $ 2000$ and we calculated the full order of the $l$-part of $h^{+}$ for all odd primes $l 10000$.

Détail

DOI : 10.4064/aa165-3-5

Keywords: class numbers real cyclotomic fields hard compute methods based discriminant bounds become useless conductor field grows methods employing leopoldts decomposition class number become hard field extension cyclic prime power why other methods have developed which approach problem different angles paper extend these methods designed real cyclotomic fields prime conductor make applicable real cyclotomic fields conductor equal product distinct odd primes main advantage method does exclude primes dividing order galois group contrast other methods applied algorithm real cyclotomic fields conductor calculated full order l part odd primes