Proof that there is always irreducible polynomial of degree $n$

Question

I do not really understand about this paragraph In this proof, what is meant by the leading term $p^n$ will dominate the remaining terms?

and also, how he conclude in the end that $\sigma(n)>0$? (Because I only see he proved that $n\sigma(n)<p^n$.)

Answer 1

He is saying that $$n\sigma(n)=p^n+\sum_{d\mid n,d<n}\mu(n/d)p^d$$ and that $$\left|\sum_{d\mid n,d<n}\mu(n/d)p^d\right|\le\sum_{e=0}^{n-1}p^e =\frac{p^n-1}{p-1}<p^n.$$ This then implies that $n\sigma(n)>0$.