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Alex Samuel Bamunoba: Cyclotomic Polynomials

Time: Wed 2019-04-10 11.00 - 12.00

Location: Room F11, KTH

Participating: Alex Samuel Bamunoba

Abstract: The rings Z and A:=Fq[T] (the polynomial ring in T over a finite field Fq) have many similarities of arithmetical interest. For example, both are integral domains, principal ideal domains with finitely many units and infinitely many prime elements. It is these similarities that make their multiplicative structures almost identical, even-though their additive structures are totally different. These similarities continue to extensions of the fraction fields of the two rings. For example, if we let k be the fraction field of A, koo be the completion of k with respect to a distinguished place oo and Coo be the completion of an algebraic closure K of koo, then the sequence of inclusions A → k → koo → Coo is analogous to Z → Q → R → C. Here, Coo is the analogue for complex numbers. In this talk, I will consider the the multiplicative group Gm of C on the one hand and the additive group Ga of Coo endowed with the Carlitz A-module structure to construct cyclotomic polynomials in the two parallel worlds, and present some results on their coefficients.