upper zenith
Given the following subring of $\mathbb C$:
$\mathbb Z \sqrt{-3} = {a+b\sqrt{-3}|a,b\in \mathbb Z}$ (it's an integral domain and principal ideal domain)
I need to prove the following:
let $c \in \mathbb Z$
a. if there exists $a,b\in \mathbb Z$ such that $a^2+3b^2=c$ then $c\equiv 0,1,3,4,9~(mod~9)$ pretty sure I managed to do that ok \
b. prove that if $p\equiv 2,5,6,8 ~(mod 9)$ odd prime integer so p is also prime in that subring we defined.
c. let p prime that satisfies section b. prove that in the ring $\mathbb Z[\sqrt -3]/ \langle p\rangle$ there is a solution to x^2 = -1
golden olive