$I=\{a+ib\in\mathbb Z[i]:5|a,5|b\}$ forms a maximal ideal of $\mathbb Z[i].$

Question

How to show that $I=\{a+ib\in\mathbb Z[i]:5|a,5|b\}$ forms a maximal ideal of $\mathbb Z[i].$

I tried to prove $\mathbb Z[i]\backslash I$ is a field could not prove it.

score 2 · Accepted Answer · answered Nov 01 '19 at 11:37

2

The two elements $x=1+2i$ and $y=1+3i$ are not in $I$, but their product $xy=-5+5i$ is. Hence $I$ doesn't even form a prime ideal of $\mathbb{Z}[i]$.

answered Nov 01 '19 at 11:37

Laassila souhayl

Can we conclude the same if 5 is replaced with 7, that is, if $I={a+ib\in\mathbb Z[i]:7|a,7|b}$? – Jave Nov 01 '19 at 18:57

1 Answers1