There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d genus 0 curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers is connected to mathematical physics, and it was not until the 1990's that it was completely solved. For example, Kontsevich determined them with a celebrated recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients. Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. A1-homotopy theory provides a framework to define analogous counts which are sensitive to the field of definition of the curves. For example, Gromov--Witten invariants in Hermitian K-theory define an arithmetically meaningful count of genus 0 plane curves. They were developed in joint work with Kass, Levine, and Solomon. This talk will completely determine the count of degree d rational curves passing through 3d-1 points over an arbitrary field of characteristic not 2 or 3. This enumeration is joint work with Erwan Brugallé and Johannes Rau.