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We discuss the role of additive polynomials and $p$-polynomials in the theoryof valued fields of positive characteristic and in their model theory. Weoutline the basic properties of rings of additive polynomials and discussproperties of valued fields of positive characteristic as modules over suchrings. We prove the existence of Frobenius-closed bases of algebraic functionfields $F|K$ in one variable and deduce that $F/K$ is a free module over thering of additive polynomials with coefficients in $K$. Finally, we prove thatevery minimal purely wild extension of a henselian valued field is generated bya $p$-polynomial.

Additive Polynomials and Their Role in the Model Theory of Valued Fields