Powering map by field characteristic is same in algebra and algebra group
Suppose is a field of characteristic equal to a prime number and is an associative algebra over that admits an algebra group . (This happens, for instance, if is nilpotent). Then, the -power map in corresponds to the -power map in , i.e.:
The proof simply involves expanding using the binomial theorem and then using characteristic . Note that we crucially use the fact that 1 commutes with to apply the binomial theorem.