What does $\varphi (n)$ denote in the context 'the class of $q$ modulo $n$ has order $\varphi (n)$'?

Question

I am currently trying to understand the following condition (found here in Corollary 48) which tells us when the $n^\text{th}$ cyclotomic polynomial $\Phi_n (x)$ (over a finite field) is irreducible:

The $n^\text{th}$ cyclotomic polynomial $\Phi_n (x)$ is irreducible in $F_q [x]$ iff the class of $q$ modulo $n$ has order $\varphi (n)$

I am hoping that someone might be able to help me understand what this means by explaining what $\varphi (n)$ denotes?

Answer 1

This refers to the Euler totient: the number of integers less than $n$ which are coprime to $n$. Your hypothesis is that $q$ has order $\phi(n)$ in $\mathbb{Z}_n^*$: equivalently, that $q$ generates $\mathbb{Z}_n^*$.