Additive Polynomials

Abstract

Let k be a field of finite characteristic p and let $\bar k$ be a fixed algebraic closure. Let $P\left( x \right) \in k\left[ x \right]$ be a polynomial in x with coefficients in k. We say that P(x) is additive if and only if $P\left( {\alpha + \beta } \right) = P\left( \alpha \right) + P\left( \beta \right)\,\,for\,\,\left\{ {\alpha ,\beta ,\alpha + \beta } \right\} \subseteq k$. We say that P(x) is absolutely additive if and only if P(x) is additive over $\bar k$.