Is this fact needed in the proof regarding division rings?

Question

I am studying the proof that non-trivial finite division rings are fields. The particular proof in question can be found here: Finite Division Rings are Fields

In the proof (between top of page 24 and before equation (1)) they make a note that there are equivalence classes of size greater than or equal to 2. This fact is indeed true, but why do we need this fact? I have skimmed the proof a few times now and do not see anywhere the fact is needed. I am asking if someone would kindly look at the proof and let me know if they see where this condition is necessary. Thanks!

Answer 1

I figured it out.

At the top of page 26, the author writes

$$x^n-1=(x^{n_k}-1)\Psi_n(x)\prod_{d\mid n,d\nmid n_k,d<n}\Psi_d(x).$$

This equation holds since $n_k$ is a proper divisor of $n$. We only know $n_k<n$ because $|A_k|>1$.

Answer 2

The key moment is when the author writes $$|R^*| = |Z^*| + \sum_{k=1}^t|A_k|$$ with $t\geqslant 1$. There the $|A_k|$ in the sum are exactly those that are strictly greater than $1$, since those which are equal to $1$ were regrouped in $|Z^*|$.

It then becomes $$q^n-1 = q-1 + \sum_{k=1}^t\frac{q^n-1}{q^{n_k}-1}$$ with $\frac{q^n-1}{q^{n_k}-1}>1$, and the fact that there are indeed such terms (in other words that $t\geqslant 1$) is crucial.