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We begin a systematic investigation of derived categories of smoothprojective toric varieties defined over an arbitrary base field. We show that,in many cases, toric varieties admit full exceptional collections. Examplesinclude all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds,the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toricvarieties associated to Weyl fans of type $A$. Our main technical tool is acompletely general Galois descent result for exceptional collections of objectson (possibly non-toric) varieties over non-closed fields.

On derived categories of arithmetic toric varieties