  The classical version of B\'ezout's Theorem gives an integer-valued count of
the intersection points of hypersurfaces in projective space over an
algebraically closed field. Using work of Kass and Wickelgren, we prove a
version of B\'ezout's Theorem over any perfect field by giving a bilinear
form-valued count of the intersection points of hypersurfaces in projective
space. Over non-algebraically closed fields, this enriched B\'ezout's Theorem
imposes a relation on the gradients of the hypersurfaces at their intersection
points. As corollaries, we obtain arithmetic-geometric versions of B\'ezout's
Theorem over the reals, rationals, and finite fields of odd characteristic.
