An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields

Abstract

In this paper we construct initial perfect quadratic formsover certain families of totally real number fields K. We assume that thenumber field K is either the maximal totally real subfield of a cyclotomicfield Q(ζn), where 3 χ n is the product of distinct odd primes p1, . . . , pk,or K = Q(√m1, . . . , √mk), where m1, . . . , mk are pairwise relativelyprime, square-free positive integers with all or all but one congruent to 1modulo 4. These perfect forms can be used to find all perfect quadraticforms of given rank (up to equivalence and proportion) over the field Kby applying the generalization of Voronoi's algorithm.