Prime ideals in $\mathbb{Z}[X], \mathbb{Q}[X], \mathbb{F}_{19}[X]$.

In fact, the implication irreducible $\implies$ prime is true for UFDs, as discussed in this question. (Moreover, assuming every element factors into irreducibles in a domain $R$, then $R$ is a UFD iff every irreducible is prime. See this page of Stacks Project for a proof.)

Hint for $J$: Divide $X^{2018} + 3X + 15$ by $X-1$ or, better yet, realize that you can find its remainder by simply plugging in $X=1$. This shows that $X^{2018} + 3X + 15 = q(X) (X-1) + 19$ for some $q(X) \in \mathbb{Z}[X]$, so $J = (X^{2018}+3X+15, X-1) = (X-1, 19)$.