Promoting my comments to an answer, as suggested by OP:
A polynomial with real coefficients is guaranteed to have a real zero if its degree is odd, and there is no guarantee of a real zero if its degree is even.
For integer $n\ge0$, $\int x^ne^{-x^2}\,dx$ is elementary if and only if $n$ is odd.
For integer $n\geq 2$, $\zeta(n)$ is known for $n$ even, in the sense that it is a rational multiple of $\pi^n$, with a simple expression for the rational in terms of Bernoulli numbers. It follows that it is transcendental for even $n$. Much less is known about the value of $\zeta(n)$ for odd $n$.
