# Computing quadratic subfields of number fields

## TL;DR

This paper presents efficient algorithms for computing quadratic and cyclic subfields of a number field, leveraging ramified primes, and significantly improves speed over existing methods for quadratic subfields.

## Contribution

It introduces a novel approach to determine ramified primes and compute all quadratic and cyclic subfields, generalizing to prime degree cyclic subfields.

## Key findings

- Efficient algorithms for quadratic subfields
- Faster than previous methods for quadratic subfields
- Generalization to cyclic subfields of prime degree

## 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, one 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 the quadratic and the 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.