[Math] Euler’s totient function and complex numbers

Question

I have this problem without a solution (yet)

Say I have a complex number and wish to calculate $\phi(a+bi)$ where $\phi(n)$ is Euler's totient function.
How would it have to be calculated?
I know that $\phi(n)$ calculates the amount of integers below n that are coprime to n. But since complex numbers form a field, what defines 'below' and what defines 'coprime'?

Thanks in advance 🙂

Wampie

Edit:

The complex numbers I am using are all Gaussian Integers

Answer 1

If $n$ is a non-zero integer, whether positive or negative, we can define $\phi(n)$ to be the order of the group of units in the ring ${\bf Z}/(n)$, where $(n)$ is the ideal of $\bf Z$ generated by $n$. Same works in the Gaussian integers: the ring ${\bf Z}[i]/I$ is finite for any non-zero ideal $I$, so $\phi(\alpha)$ can be defined as the order of the group of units in ${\bf Z}[i]/(\alpha)$.