prop_quadratic :: Nimber -> Nimber -> Bool
prop_quadratic p q =
  let (x, y) = solveQuadratic p q
   in sqr x + p * x + q == 0
        && sqr y + p * y + q == 0

    solveQuadratic,

-- | @'solveQuadratic' p q@ returns the solutions to the equation \(X^2 + px + q = 0\).
solveQuadratic :: Nimber -> Nimber -> (Nimber, Nimber)
solveQuadratic 0 q = (sqrt q, sqrt q)
solveQuadratic p q = let x = p * artinSchreierRoot (q / sqr p) in (x, x + p)