Computing Associated Orders and Galois Generating Elements of Unit Lattices Original Research Article

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