On computing subfields. A detailed description of the algorithm

Let ℚ(α) be an algebraic number field given by the minimal polynomial f of α. We want to determine all subfields ℚ(β) ⊂ Q(α) of given degree. It is convenient to describe each subfield by a pair (g, h) ∈ Z [t] x ℚ[t] such that g is the minimal polynomial of β = h(α) . There is a bijection between the block systems of the Galois group of f and the subfields of ℚ(α). These block systems are computed using cyclic subgroups of the Galois group which we get from the Dedekind criterion. When a block system is known we compute the corresponding subfield using p- adic methods. We give a detailed description for all parts of the algorithm.