Additive polynomial ($L(X+Y)=L(X)+L(Y)$)

Over a field of characteristic $p$, additive polynomials are sums of $p$-power monomials, $L(X)={\sum }_i{a}_i{X}^{p^i}$. They act linearly over the prime field. The identity follows from the Frobenius endomorphism; the converse follows by comparing mixed binomial coefficients. Their zero sets are additive groups, and a nonzero linear coefficient makes all roots simple.