On page 4 of Arithmetic of Elliptic Curves, Silverman defines the dimension of a variety $V$ (defined over $K$) as the transcendence degree of the function field $\overline{K}(V)$ over $\overline{K}$.
Why do we need to go up to the algebraic closure? I've been trying to think of an example where the transcendence degree would differ between $K(V)/K$ and $\overline{K}(V)/\overline{K}$, but I'm surprised adding algebraic elements affects it at all.