  We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$
polynomials whose resultant equals $1$. We relate the topology of these
algebraic varieties to their geometry and arithmetic. In particular, we compute
their \'{e}tale cohomology, the associated eigenvalues of Frobenius, and the
cardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ are
coprime, we show that the \'etale cohomology of
$\mathcal{R}_{n/\bar{\mathbb{F}}_q}$ is pure, and of Tate type if and only if
$q\equiv 1$ mod $n$. We also deduce the values of these invariants for the
finite field counterparts of the moduli spaces $\mathcal{M}_n$ of $SU(2)$
monopoles of charge $n$ in $\mathbb{R}^3$, and the associated moduli space
$X_n$ of strongly centered monopoles.
  An appendix by Cazanave gives an alternative and elementary computation of
the point counts.
