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We consider the moduli space $\mathcal{R}_n$ of pairs of monic, degree $n$polynomials whose resultant equals $1$. We relate the topology of thesealgebraic varieties to their geometry and arithmetic. In particular, we computetheir \'{e}tale cohomology, the associated eigenvalues of Frobenius, and thecardinality of their set of $\mathbb{F}_q$-points. When $q$ and $n$ arecoprime, 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 thefinite 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 ofthe point counts.

Topology and arithmetic of resultants, II: the resultant $=1$ hypersurface (with an appendix by C. Cazanave)