How can you show that 3 is irreducible in Z[(-5^(1/2)]?
You write $\displaystyle 3 = (a+bi\sqrt{5})(c+di\sqrt{5})$.
Expand the RHS and compare coefficients.
Now argue that $\displaystyle a=\pm 1, b=0$ or $\displaystyle c=\pm 1, d=0$.
That would $\displaystyle 3$ cannot be factored properly and non-trivially.