3 different prime ideals in Z[x] 
find 3 different prime ideals in $Z[x]$, $I,J,K$ such that $I\subset J\subset K$.  

have no clue where to start from.
 A: Hint 1. Since $\mathbb{Z}[x]$ is an integral domain, there is a very small prime ideal (in fact, it has as few elements as any ideal has any right to have).
Hint 2. There is a one-to-one, inclusion preserving correspondence between the ideals of $R/I$ and the ideals of $R$ that contain $I$. This correspondence preserves primality. So, perhaps you can find some proper prime ideal $P$ such that $\mathbb{Z}[x]/P$ (which must be an integral domain) has a proper prime ideal as well? Then you could "lift" it back to $\mathbb{Z}[x]$, and that would give you your $J$ and $K$. Then use the first hint for your $I$.
A: HINT $\ $ Factoring out a prime $\rm\:p \ne 0\:$ reduces it to finding a chain of two prime ideals in $\rm\:\mathbb F_p[x]\:.$
A similar reduction tackles lhf's question in the comments - see Theorem 37 below from Kaplansky's Commutative Rings. (Theorem 34 is the bijective order-preserving correspondence between prime ideals in a localization $\rm\:R_S\:$ and all prime ideal in $\rm\:R\:$ disjoint from $\rm\:S\:$). 

