Groups of unimodular circulants

Abstract

A me thod to determine a basis of the group of rational integral sy mmetri c posi ti ve definite uni· modular IlXII c irculants for any II. is presented . Th is method uses tllP co rrespondence between unin.vdular eirculants and units of the algebraic number field R(D, where ~ is a primitive nth root of unity. Know n resu lt s are used to obta in gene rato rs of ce rtain ape riodic subgroups of the abe li an, finitely generated gro up of un it s in RU;l. The co rres ponde nce, then , ~) ruduces the desired bas is eJements. The number of bas is elements for each II is proved to be r ~J + 1o-,,(n), where o-o(n) is the numbe)" uf posi ti ve diviso rs of n . In addition, an upper bound for t~1e number of congruence c lasses of these circulant s is obta ined , where co ngrue nce is re lative to rationa l symmetri c unimodular Ilxn circulant s.