On an extension of Niven's theorem

Abstract

For a given rational number r, a classical theorem of Niven asserts that if cos ⁡ ( rπ ) is rational, then cos ⁡ ( rπ ) ∈ { 0 , ± 1 , ± 1 / 2 } . In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that.