MathNet 2010 — Part 03 Question 73

Let $a$, $b$, $c$ be integers such that $b$ is even. Suppose the equation $x^3 + a x^2 + b x + c = 0$ has roots $\alpha$, $\beta$, $\gamma$ such that $\alpha^2 = \beta + \gamma$. Prove that $\alpha$ is an integer and $\beta \neq \gamma$.