As the title suggests, I want to have Mathematica resolve
Element[p, Primes] PolynomialMod[(1+x)^p, p]
to $1+x^p$. I have a massive symbolic function I would like to reduce in this fashion, and it involves many terms of the form above for two arbitrary primes $p$ and $q$. I can do a replace all with the reductions above, but that'll involve going through the pages of output and manually finding things to replace. I was hoping for something more automatic and elegant. Also, is it possible to have the above reduction occur within a square root?