Determine number of divisors of $x^n -1$ of degree $k$. Why are they such a number?

The field extension $?_{32} / ?_2$ can be built as the smallest splitting field of $x^{32}-x$ (it is the standard proof of existence for a field of order $p^m$). So $x^{31}-1 = (x-α_1)...(x-α_{31})$ with $?_{32} - {0} = {α_1,...,α_{31}}$, and any divisor of $x^{31}-1$ of degree $k$ can be obtained by selecting $k$ factors from $(x-α_1)...(x-α_{31})$ (order of selection irrelevant).