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Let F be a finite extension of ℚ. Using the mod p Satake transform, we define what it means for an irreducible admissible smooth representation of an F-split p-adic reductive group over $\overline{ \mathbb{F}}_{p}$ to be supersingular. We then give the classification of irreducible admissible smooth GL( F)-representations over $\overline{ \mathbb{F}}_{p}$ in terms of supersingular representations. As a consequence we deduce that supersingular is the same as supercuspidal. These results generalise the work of Barthel-Livné for n=2. For general split reductive groups we obtain similar results under stronger hypotheses. [ABSTRACT FROM AUTHOR]