  Let $F$ be a finite type surface and $\zeta$ a complex root of unity. The
Kauffman bracket skein algebra $K_{\zeta}(F)$ is an important object in both
classical and quantum topology as it has relations to the character variety,
the Teichm\"uller space, the Jones polynomial, and the
Witten-Reshetikhin-Turaev Topological Quantum Field Theories. We compute the
rank and trace of $K_{\zeta}(F)$ over its center, and we extend a theorem of
Frohman and Kania-Bartoszynska which says the skein algebra has a splitting
coming from two pants decompositions of $F$.
