Abstract
LetMbe either the field of rationals Q or a quadratic imaginary number field. We denote byNan abelian number field extension ofM, which is supposed to be real ifM=Q. We putΓ=Gal(N/M) and letUNbe the unit lattice ofN. We describe an algorithm that first computes the associated order AN/M={λ∈Q[Γ]/(∑γ∈Γγ)∣λ(UN)⊆UN} ofUNand then answers the question of whether or notUNis a free, rank one module over AN/M. If the answer is positive a generator also is computed. In particular, we can computationally decide whether a Minkowski unit exists.
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