Product of numbers $\pm\sqrt{1}\pm\sqrt{2}\pm\cdots\pm\sqrt{n}$ is integer
Hint: Let $P_n(x)$ be your polynomial. Then show $P_{n+1}(x)=P_n(x-\sqrt{n+1})P_n(x+\sqrt{n+1})$, and show inductively that $P_n(x)$ always has only integer coefficients.
Hint: Let $P_n(x)$ be your polynomial. Then show $P_{n+1}(x)=P_n(x-\sqrt{n+1})P_n(x+\sqrt{n+1})$, and show inductively that $P_n(x)$ always has only integer coefficients.