Additive Polynomial -- from Wolfram MathWorld

Let be a field of finite characteristic . Then a polynomial is said to be additive iff for . For example, is additive for , since

A more interesting class of additive polynomials known as absolutely additive polynomials are defined on an algebraic closure of . For example, for any such , is an absolutely additive polynomial, since , for , ..., . The polynomial is also absolutely additive.

Let the ring of polynomials spanned by linear combinations of be denoted $k{tau_p}$ . If , then $k{tau_p}$ is not commutative.

Not all additive polynomials are in $k{tau_p}$ . In particular, if is an infinite field, then a polynomial is additive iff $P(x) in k{tau_p}$ . For be a finite field of characteristic , the set of absolutely additive polynomials over equals $k{tau_p}$ , so the qualification "absolutely" can be dropped and the term "additive" alone can be used to refer to an element of $k{tau_p}$ .

If is a fixed power and , then is a ring of polynomials in . Moreover, if , then for all . In this case, is said to be a -linear polynomial.

The fundamental theorem of additive polynomials states that if is a separable polynomial and ${omega_1,...,omega_n} subset k$ is the set of its roots, then is additive iff if ${omega_1,...,omega_n}$ is a subgroup.