  We study toric varieties over an arbitrary field with an emphasis on toric
surfaces in the Merkurjev-Panin motivic category of "K-motives". We explore the
decomposition of certain toric varieties as K-motives into products of central
simple algebras, the geometric and topological information encoded in these
central simple algebras, and the relationship between the decomposition of the
K-motives and the semiorthogonal decomposition of the derived categories. We
obtain the information mentioned above for toric surfaces by explicitly
classifying all minimal smooth projective toric surfaces using toric geometry.
