 Nonprincipal ideal generated by a prime and an indeterminate

For a prime number $p$, the ideal $(p,X)$ in the polynomial ring $\mathbb{Z}[X]$ is proper: the constant coefficient of every member is divisible by $p$. If $(p,X)=(g)$, then $g$ divides the nonzero constant $p$, so the additivity of polynomial degree over an integral domain makes $g$ constant. Since $g$ also divides $X$, it must divide its leading coefficient $1$; thus $g=\pm 1$. This would make the ideal the whole ring, a contradiction. Therefore $\mathbb{Z}[X]$ is not a principal ideal domain, and hence not a Euclidean domain.