Shortest nonzero lattice points in totally real multi-quadratic number fields and applications

Let [Formula: see text] be a multi-quadratic totally real number field. Let [Formula: see text] denote its distinct embeddings. Given [Formula: see text] we give an explicit formula for [Formula: see text] and [Formula: see text] where [Formula: see text] Let [Formula: see text] be a fractional ideal in [Formula: see text] and [Formula: see text] The set of shortest nonzero lattice points for [Formula: see text] is given by [Formula: see text] We provide shortest nonzero lattice points for [Form