Solution:

We shall prove that Elitza has a winning strategy. If the polynomial is $a_{0} x^{4} + a_{1} x^{3} + a_{2} x^{2} + a_{3} x + a_{4}$ and Aleksander writes $a_{0}$, $a_{1}$, $a_{2}$ or $a_{3}$, then Elitza writes respectively $a_{1} = a_{0}$, $a_{0} = a_{1}$, $a_{3} = a_{2}$ or $a_{2} = a_{3}$; if he writes $a_{4}$, she writes $a_{1} = 1$.

In a similar way Elitza is able to get $a_{1} \leq a_{0}$ and $a_{3} \leq a_{2}$ after her second move. Suppose that the polynomial obtained has an integer root $-y$. Then $y \geq 1$ and hence $a_{4} = y^{3}(a_{1} - a_{0} y) + a_{3} - a_{2} y \leq 0$, which is a contradiction.