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The pentagram map, introduced by Schwartz in 1992, is a dynamical system onthe moduli space of polygons in the projective plane. Its real and complexdynamics have been explored in detail. We study the pentagram map over anarbitrary algebraically closed field of characteristic not equal to 2. We provethat the pentagram map on twisted polygons is a discrete integrable system, inthe sense of algebraic complete integrability: the pentagram map is birationalto a self-map of a family of abelian varieties. This generalizes Soloviev'sproof of complex integrability. In the course of the proof, we construct themoduli space of twisted $n$-gons, derive formulas for the pentagram map, andcalculate the Lax representation by characteristic-independent methods.

The Algebraic Dynamics of the Pentagram Map