Computing quadratic subfields of number fields

Abstract

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, we can try to determine them by using class field theory. For this, it is necessary to know the ramified primes. We show that the ramified primes of the subfield can be computed efficiently. Using this information, we give algorithms to determine all quadratic and cyclic cubic subfields of the initial field. The approach generalises to cyclic subfields of prime degree. In the case of quadratic subfields, our approach is much faster than other methods.