Let $\displaystyle R=Z[\sqrt{-5}]$

a) Show that:

i) $\displaystyle \langle 2 \rangle = \langle 2, 1 + \sqrt{-5} \rangle^2$

ii) $\displaystyle \langle 3 \rangle = \langle 3, 1 + \sqrt{-5} \rangle \langle 3, 1 - \sqrt{-5} \rangle $

b) Prove that all ideals that appear on the right in equations of part a) are prime using that $\displaystyle R/\langle 2 \rangle$ have four elements.