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Feb 23, 2016 at 7:04 comment added KConrad Yes, such a formula holds. Take a proof of the classical identity in $\mathbf Z$ and try to carry it over. Did you attempt to do that? If not, you should. By the way, in the formula you ask about you should write that $d$ runs over monic factors of $f$.
Feb 23, 2016 at 0:17 comment added Lev Borisov Well, if $f$ is irreducible, then you have the usual proof of Fermat's Little Theorem apply to the image of $a$ in $F[x]/<f>$, so it should be OK.
Feb 23, 2016 at 0:09 history asked user76513 CC BY-SA 3.0