AUSTRIAN-POLISH 1992 Problem 4

Let $k$ be a positive integer and $u, v$ be real numbers. Consider $P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c$ . (a) For $k = 2$ prove that if $a, b, c$ are rational then so is $uv$ . (b) Is that also true for $k = 3$ ?