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2008 | OriginalPaper | Chapter
An Explicit Construction of Initial Perfect Quadratic Forms over Some Families of Totally Real Number Fields
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In this paper we construct initial perfect quadratic forms over certain families of totally real number fields
. We assume that the number field
is either the maximal totally real subfield of a cyclotomic field
, where
${3 \not |\, n}$
is the product of distinct odd primes
p
1
,...,
p
k
, or
, 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
by applying the generalization of Voronoi’s algorithm.