Question 117
Let $N$ be the number of positive integers with 10 digits $\overline{d_{9} d_{8} \cdots d_{1} d_{0}}$ in base 10 (where $0 \leq d_{i} \leq 9$ for all $i$ and $d_{9} \gt 0$) such that the polynomial
$$
d_{9} x^{9}+d_{8} x^{8}+\cdots+d_{1} x+d_{0}
$$
is irreducible in $\mathbb{Q}$. Prove that $N$ is even.
(A polynomial is irreducible in $\mathbb{Q}$ if it cannot be factored into two non-constant polynomials with rational coefficients.)
$$
d_{9} x^{9}+d_{8} x^{8}+\cdots+d_{1} x+d_{0}
$$
is irreducible in $\mathbb{Q}$. Prove that $N$ is even.
(A polynomial is irreducible in $\mathbb{Q}$ if it cannot be factored into two non-constant polynomials with rational coefficients.)