  In this paper, the nilspace approach to higher-order Fourier analysis is
developed in the setting of vector spaces over a prime field $\mathbb{F}_p$,
with applications mainly in ergodic theory. A key requisite for this
development is to identify a class of nilspaces adequate for this setting. We
introduce such a class, whose members we call $p$-homogeneous nilspaces. One of
our main results characterizes these objects in terms of a simple algebraic
property. We then prove various further results on these nilspaces, leading to
a structure theorem describing every finite $p$-homogeneous nilspace as the
image, under a nilspace fibration, of a member of a simple family of filtered
finite abelian $p$-groups. The applications include a description of the
Host-Kra factors of ergodic $\mathbb{F}_p^\omega$-systems as $p$-homogeneous
nilspace systems. This enables the analysis of these factors to be reduced to
the study of such nilspace systems, with central questions on the factors thus
becoming purely algebraic problems on finite nilspaces. We illustrate this
approach by proving that for $k\leq p+1$ the $k$-th Host-Kra factor is an
Abramov system of order $\leq k$, extending a result of Bergelson-Tao-Ziegler
that holds for $k< p$. We illustrate the utility of $p$-homogeneous nilspaces
also by showing that the above-mentioned structure theorem yields a new proof
of the Tao-Ziegler inverse theorem for Gowers norms on $\mathbb{F}_p^n$.

Dynamical Systems;Combinatorics