Cyclic division algebras with non-norm elements

In this paper, we construct some cyclic division algebras (K/F, σ,γ). We obtain a necessary and sufficient condition of a non-norm element γ provided that F = ℚ and K is a subfield of a cyclotomic field ℚ(ζ_p^u), where p is a prime and ζ_p^u is a p^uth primitive root of unity. As an application for space time block codes, we also construct cyclic division algebras (K/F, σ, γ), where F = ℚ(i) i=-1, K is a subfield of ℚ(ζ_4p^u) or ℚ (ζ4p^u ₁p^v₂), and γ = 1+i. Moreover, we describe all cyclic division algebras (K/F,σ,γ) such that F= ℚ(i), K is a subfield of (ζ4p^u₁p^v₂)and γ=1+i, where [K : F] = φ( p^u₁p^v ₂)/d, d=2 or 4, φ is the Euler totient function, and p1, p2≤ 100 are distinct odd primes.