Example 9 If the discriminant of $x^{2}+2 k x+2 t-1=0$ is to be determined ().
(A) is an integer
(B) is an integer, but it cannot be determined whether it is odd or even
(C) is an odd number
(D) is an even number
Solution: \| $\alpha+\beta=-2k, \beta=-2k-\alpha$ are integers, and $\alpha \beta=2t-1$ is odd,
$\therefore \alpha, \beta$ are both odd.
Therefore, the answer is (C).