TITLE: $\mathbb Z[i]$ and the ideal $(5)$
QUESTION [5 upvotes]: Consider $\mathbb Z[i]$ the ring of Gaussian integers and its ideal $J=(5)$. Show that
$\mathbb Z[i]/J \cong \mathbb Z_5 \oplus \mathbb Z_5$ as rings.

REPLY [10 votes]: Hint $\ \Bbb Z[i]/5 \cong \Bbb Z[x]/(5,x^2\!+1) = \Bbb Z[x]/(5,x^2\!-4)\!\! \overset{\rm CRT}\cong\! \Bbb Z_5[x]/(x\!-\!2) \times \Bbb Z_5[x]/(x\!+\!2) \cong \Bbb Z_5\!\! \times\! \Bbb Z_5\!\!$