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Motivated by the search for a logic for polynomial time, we study rank logic (FPR) which extends fixed-point logic with counting (FPC) by operators that determine the rank of matrices over finite fields. While FPR can express most of the known queries that separate FPC from PTIME, nearly nothing was known about the limitations of its expressive power. In our first main result we show that the extensions of FPC by rank operators over different prime fields are incomparable. This solves an openarXiv:1503.05423v1 fatcat:coz4qftg65go7dcttf6t3guwwa