Fields with Additive identity powers

Question

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 $a\neq b$.

Answer 1

Try showing that $kx=k$ for any $x$ in the field using the field axioms.

Solution below if you are stuck.

$kx=(k+k)x=kx+kx$ so subtracting $kx$ from both sides we get $k=kx$.