  Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let
$V$ be a linear representation of $G$. Geometric invariant theory involves the
study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation
between these invariants and the $G$-orbits on $V$, usually under the
hypothesis that the base field $k$ is algebraically closed. In favorable cases,
one can determine the geometric quotient $V//G = Spec(Sym(V^*))^G$ and can
identify certain fibers of the morphism $V \rightarrow V/G$ with certain
$G$-orbits on $V$. In this paper, we study the analogous problem when $k$ is
not algebraically closed. The additional complexity that arises in the orbit
picture in this scenario is what we refer to as arithmetic invariant theory. We
illustrate some of the issues that arise by considering the regular semi-simple
orbits--i.e., the closed orbits whose stabilizers have minimal dimension--in
three arithmetically rich representations of the split odd special orthogonal
group $G = SO_{2n+1}$.
