On the Computation of the Norm-Euclidean Minimum of Algebraic Number Fields
Michele Elia, J. Carmelo Interlando · PORTO Publications Open Repository TOrino (Politecnico di Torino) · 2009
Let F be an algebraic number field whose group of units has rank ≥ 1. The conjecture that the norm-Euclidean minimum M (F) is a rational number is affirmatively settled. It is proved that M (F) is lower bounded by the inverse of the smallest norm of all nonzero prime OF-ideals. Furthermore, when F/Q is a normal extension, the numerator and denominator of M (F) lie within finite sets of integers that can be explicitly calculated. As an application, it is proved that the known lower bounds of M (F) for the cyclotomic field Q(ζ5) and the cyclic cubic fields of discriminants 103 2 , 109 2 , 117 2 , and 157 2 are the actual values of M (F).