Thesis Defence: On the Length of Cyclic Algebras and Corrigenda to Articles

Amaury De Burgos, supervised by Dr. Javad Tavakoli, will defend their thesis titled “On the Length of Cyclic Algebras and Corrigenda to Articles” in partial fulfillment of the requirements for the degree of Master of Science in Mathematics.

An abstract for Amaury De Burgos’ thesis is included below.

Defences are open to all members of the campus community as well as the general public. Please email javad.tavakoli@ubc.ca to receive the Zoom link for this defence.

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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 (i.e. the Galois group is a cyclic group). This thesis provides corrigenda to two articles whose results depend on and/or concern cyclic algebras; “Untying knots in 4D and Wedderburn’s theorem” [Ni₂] and “Lengths of cyclic algebras and commutative subalgebras of quaternion matrices” [Mi]. In particular, we disprove Theorem 1.1 of the former article and Theorem 2.8 of the latter through counterexamples.

By finding an inert rational prime, we prove the maximal real subfield Q(ω₂ᵏ + ω⁻¹₂ᵏ) of the 2ᵏ -th cyclotomic field (with 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. From this extension, we construct an infinite family of cyclic division algebras and give a lower bound on the lengths of its members (i.e. the length of the longest chain of linear subspaces arising from a generating set for the algebra). Lastly, we tensor the members of this family with Q(i) to produce fully diverse linear space-time block codes with non-vanishing determinant for 2ᵏ⁻² transmit antennas.