Toric surfaces over arbitrary fields

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