Euler totient function for Gaussian integers

Question

Actually this not a question but just an observation followed by a small question. For primes $p$, $\phi(p)=p-1$, for general $n$, $\phi(n)<n$ since $\phi(n)$ is the order of the group $\mathbb{Z}_n^*$ of units of $\mathbb{Z}_n$.

But for $n=1$, $\phi(1)=1$ because $\mathbb{Z}_1^*=\{1\}$ hence $\phi(1)$ is not strictly less than $1$.

Now a small question I have is, how is an analogous Euler totient function defined for $\mathbb{Z}[i]$ the Gaussian integers? Or more generally algebraic integers?

Answer 1

You can define $\varphi(a + bi)$ to be the size of the group of units of the quotient $\mathbb{Z}[i]/(a + bi)$ in exactly the same way as for integers. We have

• $\varphi(mn) = \varphi(m) \varphi(n)$ if $\gcd(m, n) = 1$
• $\varphi(p) = N(p) - 1 = a^2 + b^2 - 1$ if $p = a + bi$ is prime (this means $a^2 + b^2$ is either $2$, an ordinary prime congruent to $1 \bmod 4$, or the square of an ordinary prime congruent to $3 \bmod 4$)
• $\varphi(p^k) = N(p)^{k-1} \varphi(p)$ if $p$ is prime

which determines $\varphi(-)$ completely. As in $\mathbb{Z}$ we have a totient theorem to the effect that if $\gcd(z, a + bi) = 1$ then $z^{\varphi(a + bi)} \equiv 1 \bmod (a + bi)$.

For a general ring of integers $\mathcal{O}_K$ we should talk about $\varphi(I)$ where $I$ is an ideal since there is the possibility of non-principal ideals; this will be the size of the group of units of $\mathcal{O}_K/I$, and then again we have a totient theorem $z^{\varphi(I)} \equiv 1 \bmod I$.