Chowla's theorem over function fields

Sarvadaman Chowla proved that if p is an odd prime, then cot(2 pi j/p) (j = 1, . . . , (p - 1)/2) are linearly independent over the field of rational numbers. We establish an analog of this result over function fields. As an application, we prove an analog of the Baker-Birch-Wirsing theorem about the non-vanishing of Dirichlet series with periodic coefficients at s = 1 in the function field setup with a parity condition.