Computing automorphisms of abelian number fields

Abstract

Let L = Q ( α ) L=\mathbb {Q}(\alpha ) be an abelian number field of degree n n . Most algorithms for computing the lattice of subfields of L L require the computation of all the conjugates of α \alpha . This is usually achieved by factoring the minimal polynomial m α ( x ) m_{\alpha }(x) of α \alpha over L L . In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α \alpha , which is based on p p -adic techniques. Given m α ( x ) m_{\alpha }(x) and a rational prime p p which does not divide the discriminant disc ⁡ ( m α ( x ) ) \operatorname {disc} (m_{\alpha }(x)) of m α ( x ) m_{\alpha }(x) , the algorithm computes the Frobenius automorphism of p p in time polynomial in the size of p p and in the size of m α ( x ) m_{\alpha }(x) . By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α \alpha .