I need an example of an irreducible number in form of a+bsqrt(p) that its norm is not prime.
Consider $\displaystyle \mathbb{Z}[i]$ and then any prime $\displaystyle p\equiv 3(\bmod 4)$ is a Gaussian prime and its norm is $\displaystyle p^2$ - not a prime.
More specifially, $\displaystyle 3\in \mathbb{Z}[i]$ and $\displaystyle N(3) = 9$ with $\displaystyle 3$ being irreducible.