Compute ZZ[sqrt(-5)]/(3)

Compute \mathbb{Z}[\sqrt{-5}]/(3).


By Corollary 9.11, |\mathbb{Z}[\sqrt{-5}]/(3)| = N((3)) = N(3) = 9.

Suppose a+b\sqrt{-5} \in \mathbb{Z}[\sqrt{-5}], and suppose further that a \equiv a_0 and b \equiv b_0 mod 3, where a_0,b_0 \in \{0,1,2\}. Then a+b\sqrt{-5} \equiv a_0+b_0\sqrt{-5} mod (3). That is, \mathbb{Z}[\sqrt{-5}]/(3) = \{\overline{a+b\sqrt{-5}} \ |\ a,b \in \{0,1,2\}\}. Since we know that this ring has 9 elements, none of these cosets are equal.

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