Would it be possible to have a field (or field-like structure) with an additive identity $k$ where $k^a\neq k^b$ for $a\neq b$?
I need this because I'm working with a field-like structure where if I have a mathematical object $s$, $s+s^2$ doesn't make sense. Therefore, If I have $k^a = k^b$, I run into a problem by doing:
$$ s+k = s \;\therefore\; s+(s^2-s^2) = s \;\therefore\; (s+s^2)-s^2 = s $$
So I fix this by adding in the axiom $s^n-s^n =k^n$, which then leads me to $k^a\neq k^b$ for $a\neq b$.