Title: On the trace of powers of Algebraic integers
Speaker: R. Thangadurai
Affiliation: Harish-Chandra Research Institute, Prayagraj
Affiliation: Harish-Chandra Research Institute, Prayagraj
Venue: Seminar Hall, KSoM
Date and Time: June 10, 2024 [Monday] at 11:00 AM
Date and Time: June 10, 2024 [Monday] at 11:00 AM
Abstract: Let 
be a non-zero algebraic integer. In this lecture, we prove an interesting characterisation for to be a root of unity, which is an extension of a classical theorem of Kronecker. Indeed, we prove that {\it if a non-zero algebraic integer is a root of unity if and only if the sequence /%5Cmathbb%7BQ%7D%7D(%5Calpha%5En)%5Cright)%5F%7Bn%5Cgeq%091%7D "\left(\mathrm{Tr}_{\mathbb{Q}(\alpha)/\mathbb{Q}}(\alpha^n)\right)_{n\geq 1}")
is bounded. Moreover, if the sequence is bounded, then it is periodic.} Thus, if a non-zero algebraic integer is not a root of unity, it is clear that the sequence is unbounded and hence we study the growth in the next result. Also, we introduce a problem of Polya and its extensions.
be a non-zero algebraic integer. In this lecture, we prove an interesting characterisation for to be a root of unity, which is an extension of a classical theorem of Kronecker. Indeed, we prove that {\it if a non-zero algebraic integer is a root of unity if and only if the sequence is bounded. Moreover, if the sequence is bounded, then it is periodic.} Thus, if a non-zero algebraic integer is not a root of unity, it is clear that the sequence is unbounded and hence we study the growth in the next result. Also, we introduce a problem of Polya and its extensions.