Niven's theorem

The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.

[3] Niven's proof of his theorem appears in his book Irrational Numbers.

Namely, Lehmer showed that for relatively prime integers k and n with n > 2, the number 2 cos(2πk/n) is an algebraic number of degree φ(n)/2, where φ denotes Euler's totient function.

Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6.

[4] In 1956, Niven extended Lehmer's result to the other trigonometric functions.