Factor ring and prime elements 
My task is to find prime elements of the ring $\mathbb Z[\sqrt{-21}]$ and describe the factor ring $\mathbb Z[\sqrt{-21}]/(2+\sqrt{-21})$.

I think that to describe factor-ring i need to find the norm of $(2+\sqrt{-21})$, which is equal to $25$. Then i need to say that factor-ring is isomorphic to original ring divided on this norm. Is this right?
And what about prime elements? I tried to do it with the Legendre symbol. Well suppose $p$ is prime number, $p = x^2 + 21y^2$, $0 <x,y < p$.
$x^2 + 21y^2 = 0\bmod p$.
$(xy^{-1})^2 = -21\bmod p$, then $(\frac{-21}{p}) = 1$. Then $(\frac{-21}{p})= (\frac{-1}{p})(\frac{7}{p})(\frac{3}{p})$.
So then i write Legendre symbol for $-1, 7, 3$. But what should i do further?
Thanks for any help.
 A: First, note that $ \mathbb{Z}[\sqrt{-21}] $ is the ring of integers in the quadratic number field $ \mathbb{Q}(\sqrt{-21}) $. You are right in saying that $ | \mathbb{Z}[\sqrt{-21}]/ (2 + \sqrt{-21}) | = |N(2+\sqrt{-21})| = 25 $. The prime $ (5) $ factorises as $$ (5) = (5, 2 + \sqrt{-21})(5, -2 + \sqrt{-21}) $$ Here, the ideal $ (5, 2 + \sqrt{-21}) $ is prime (its norm is equal to $ 5 $), hence maximal (We're in Dedekind Domains). So we get, using the isomorphism theorem, $$ \frac{\mathbb{Z}[\sqrt{-21}]/ (2 + \sqrt{-21})}{(5, 2 + \sqrt{-21})/(2 + \sqrt{-21})} \cong \mathbb{Z}[\sqrt{-21}]/ (5, 2 + \sqrt{-21}) \cong \mathbb{Z}/5\mathbb{Z}$$ To see that $ \mathbb{Z}[\sqrt{-21}]/ (2 + \sqrt{-21}) $ is not $ \mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z} $, observe that the coset, $ 1 + (2 + \sqrt{-21}) $ has order $ 25 $, hence $$ \mathbb{Z}[\sqrt{-21}]/ (2 + \sqrt{-21}) \cong \mathbb{Z}/25\mathbb{Z} $$
A: Here's a direct solution using the Third Isomorphism Theorem:
\begin{align*}
\frac{\mathbb{Z}[\sqrt{-21}]}{(2 + \sqrt{-21})} &\cong \frac{\mathbb{Z}[x]}{(2 + x, x^2 + 21)} \cong \frac{\mathbb{Z}}{((-2)^2 + 21)} \cong \frac{\mathbb{Z}}{(25)}
\end{align*}
where the second isomorphism is induced by the evaluation map $x \mapsto -2$.
