Regarding Gaussian integers and primitive roots.

Question

Can modular arithmetic be set up using gaussian integers instead of (non-complex) integers? If so is there an analogue of 'primitive roots' with Gaussian integers?

Answer 1

For primes $p=1 \pmod 4$ the number $i$ reduces either of two integers, and thus any $a+bi$ reduces to some number $\mod p$. Primitive roots work as normal.

For primes $p=3 \pmod 4$, there is no primitive root, and one can not reduce past a modulo class $a+bi \pmod p$.

The same is true for eisenstein integers, except the modulo classes are $p=1 \pmod 6$ and $p=5 \pmod 6$.

These integer systems are closely related to the Pell systems (of the form $(\sqrt{n+2}+\sqrt{n-2})/2$), being the solutions for n=0, n=1 respectively.

For the values n=3 and greater, one can find primitive roots for both the lower and upper primes. These have a period of $p-1$ and $p+1$ respectively.