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Abstract

A finite-dimensional central simple algebra A over a field F is called cyclic if it contains a strictly maximal subfield K such that K/F is a cyclic field extension. This thesis provides corrigenda to two articles whose results are related to cyclic algebras; "Untying knots in 4D and Wedderburn's theorem" [Ni₂], and "Lengths of cyclic algebras and commutative subalgebras of quaternion matrices" [Mi]. We correct the proof of Lemma 3.1 of the former article and provide counterexamples to Theorem 1.1 of the former article and Lemma 2.6 of the latter article. By characterizing the inert rational primes, we prove the maximal real subfield Q(ω₂ĸ}+ω₂ĸˉ¹) of the 2ᵏ-th cyclotomic field for k > 2 is a cyclic extension of Q. We also give an explicit description of all the generators of the Galois group in terms of Chebyshev polynomials of the first kind. From this extension, we construct an infinite family of cyclic division algebras and give a lower bound of the lengths of its members. Lastly, we tensor members of a subfamily with Q(i) to produce fully diverse linear space-time block codes with non-vanishing determinant for wireless communication systems with 2ᵏˉ² transmit antennas.