abelian varieties - Are Jacobians principally polarized over non-algebraically closed fields?

Incidentally, as I posted this question someone who knew the answer wandered into my office.

The map $M_g to A_g$ factors through the moduli space $tau_g$ of pairs (A,P,L) where A is an abelian variety, P is an A torsor, and L is an ample line bundle on P which is geometrically a principal polarization. The map $M_g to tau_g$ is given by $C mapsto (Pic_0, Pic_{g-1}, L(theta))$, where the theta divisor on $Pic_{g-1}$ is given by the image of $C^{g-1}$.

To construct the map $tau_g to A_g$, note that $Pic_0(A) cong Pic_0(P)$, so that L indeed gives a map $A to A^{vee}$ given by $a mapsto t^*_aL otimes L^{-1}$.

The point is one doesn't need to descend the theta divisor. The reference to this is 5.1 of Martin Olsson's book Compactifying moduli spaces of abelian varieties.