ABSTRACT. A formula is obtained for the rank of the 2-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the 2-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a Z 2 -matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both 2-class and class group Z 2 ðZ 2 are determined. The final results assume the completeness of D. A. Buell's list of imaginary fields with small class numbers.
This article concerns the minimal knotting number for several types of lattices, including the face-centered cubic lattice (fcc), two variations of the body-centered cubic lattice (bcc-14 and bcc-8), and simple-hexagonal lattices (sh). We find, through the use of a computer algorithm, that the minimal knotting number in sh is 20, in fcc is 15, in bcc-14 is 13, and bcc-8 is 18.
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