What is the meaning of the polynomial is 'absolutely additive?'
More precisely, let $k$ be a field with characteristic $p$ and $\overline{k}$ be its algebraic closure. Goss' book 'basic structure of function field arithmetic' explain the additive polynomial in the first page,
'We say that $P(x)$ is additive if and only if $P(a+b)=P(a)+P(b)$ for $\{a,b,a+b\} \subset k$. We say that $P(x)$ is absolutely additive if and only if $P(x)$ is additive over $\overline{k}$.
Is it right that the 'absolutely additive' means $P(a+b)=P(a)+P(b)$ for $a,b \in \overline{k}$?