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

摘要

In this paper we construct initial perfect quadratic forms over certain families of totally real number fields IK. We assume that the number field IK is either the maximal totally real subfield of a cyclotomic field Q(ζ_n), where 3 is the product of distinct odd primes p_1,... ,p_k, or IK = Q(m_1~(1/2),...,m_k~(1/2)), where m_1,..., m_k are pairwise relatively prime, square-free positive integers with all or all but one congruent to 1 modulo 4. These perfect forms can be used to find all perfect quadratic forms of given rank (up to equivalence and proportion) over the field IK by applying the generalization of Voronoi's algorithm.