Cyclic Division Algebras with Non-norm Elements

In this paper, we construct some cyclic division algebras (K/F,sigma,gamma). We obtain a necessary and sufficient condition of a non-norm element gamma provided that F = Q and K is a subfield of a cyclotomic field Q(zeta p(u)), where pisaprime and zeta p(u) is a p(u) th primitive root of unity. As an application for space time block codes, we alsoconstruct cyclic division algebras (K/F, sigma, gamma), where F = Q(i), i root-1, K is a subfield of Q (zeta 4p(u)) or Q (zeta 4p(1)(u)p(2)(v)) , and gamma = 1+i. Moreover, we describe all cyclic division algebras (K/F, sigma, gamma) such that F = Q(i), K is a subfield of L = Q(zeta 4p(1)(u)p(2)(v)) and gamma = 1+i, where [K : F] = phi(p(1)(u)p(2)(v))/d, d = 2 or 4, phi is the Euler totient function, and p(1), p(2) <= 100 are distinct odd primes.