2025/08/08 by Keilthy, Adam, Ruairí, Ailbhe Ní
#11R04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2508.06415
Niven's Theorem asserts that \cos(rπ)|r∈ ℚ\∩ℚ = \0, ± 1, ±(1)/(2)\. This paper uses elementary methods to classify all elements in the sets \cosn(rπ)|r∈ ℚ, n ∈ ℕ\∩ℚ and \sinn(rπ)|r∈ ℚ, n ∈ ℕ\∩ℚ. Using some algebraic number theory, we extend this to a classification of all elements in \tann(rπ)|r∈ ℚ, n ∈ ℕ\∩ℚ. Finally, we present a short Galois theoretic argument to provide a more conceptual understanding of the results.