On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial

Recently, Blanco-Chacón proved the equivalence between the Ring Learning With Errors and Polynomial Learning With Errors problems for some families of cyclotomic number fields by giving some upper bounds for the condition number Cond(Vn) of the Vandermonde matrix Vn associated to the nth cyclotomic polynomial. We prove some results on the singular values of Vn and, in particular, we determine Cond(Vn) for n = 2kpℓ, where k, ℓ ≥ 0 are integers and p is an odd prime number.

ON AUTOMORPHISMS OF EXTREMAL EVEN UNIMODULAR LATTICES

The automorphism groups of the three known extremal even unimodular lattices of dimension 48 and the one of dimension 72 are determined using the classification of finite simple groups. Restrictions on the possible automorphisms of 48-dimensional extremal lattices are obtained. We classify all extremal lattices of dimension 48 having an automorphism of order m with φ(m) > 24. In particular the lattice P48nis the unique extremal 48-dimensional lattice that arises as an ideal lattice over a cyclotomic number field.

RAY CLASS GROUPS OF QUADRATIC AND CYCLOTOMIC FIELDS

This paper studies Galois extensions over real quadratic number fields or cyclotomic number fields ramified only at one prime. In both cases, the ray class groups are computed, and they give restrictions on the finite groups that can occur as such Galois groups. Let [Formula: see text] be a real quadratic number field with a prime P lying above p in ℚ. If p splits in K/ℚ and p does not divide the big class number of K, then any pro-p extension of K ramified only at P is finite cyclic. If p is inert in K/ℚ, then there exist infinite extensions of K ramified only at P. Furthermore, for big enough integer k, the ray class field (mod Pk+1) is obtained from the ray class field (mod Pk) by adjoining ζpk+1. In the case of a regular cyclotomic number field K = ℚ(ζp), the explicit structure of ray class groups (mod Pk) is given for any positive integer k, where P is the unique prime in K above p.