Past exam of the algebra and analysis I for mathematics and philosophy course of the University of Oxford / 2013 / Algebra / 3 / c / Solution

For the Gaussian integers, use the Gaussian normTo identify the ideal, note that $a^2+b^2$ is even if and only if $a,b$ have the same parity. AlsoThis quotient is a Gaussian integer precisely when $a,b$ have the same parity. Therefore the prescribed set consists exactly of the multiples of $1+i$, givingAny associate of $1+i$ would generate the same principal ideal.