Algebra Seminar

In this talk we present algorithms for computing automorphisms and subfields of an algebraic number field K. The computations are done in unramified p-adic extensions without knowing the Galois group of (the splitting field of) K. For the subfield computation we use the fact that the lattice of block systems of the Galois group of K is isomorphic to the lattice of subfields. We present an algorithm for computing Frobenius automorphisms of normal number fields.

Using these algorithms we describe an explicit method to compute automorphisms and subfields of algebraic function fields. In the number field case it is possible to compute automorphisms and subfields of fields up to degree 60. In the abelian case it is possible to compute automorphisms for fields up to degree 200.