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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 :)



The complex numbers I am using are all Gaussian Integers

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The complex numbers don't have an order relation, and just the same, since the comlex numbers form a field, we can say every nonzero number divides every complex number (that is, there is no notion of coprime). If you define some relation, say the norm of complex numbers or something more clever, you might be able to expand the definition to the Gaussian integers. – Clayton Feb 2 '13 at 13:59
To add on to what Julien said, a more tractable problem is "How would I generalize $\phi$ to the rationals?" Without thinking too hard, my intuition is that $\phi$ is interesting because of the ideal structure on $\mathbb{Z}$, and that considering rings with less interesting ideal structures would make $\phi$ less interesting. – Eric Stucky Feb 2 '13 at 14:02
For gaussian integers, the relation $z_1|z_2$ is only true for some choices of $z_1,z_2$ so that gcd could be defined. Then maybe $\phi(z)$ could mean the number of $w$ of norm at most the norm of $z$ which are coprime to $z$ in the sense that $gcd(z,w)$ is a unit, i.e. is one of $1,i,-1,-i$. – coffeemath Feb 2 '13 at 14:22
2… – user58512 Feb 2 '13 at 14:32
up vote 4 down vote accepted

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 ${\bf Z}[i]/(\alpha)$.

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