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 .
If ,
then
is not commutative.

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

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 is the set of its roots, then
is additive iff if is a subgroup.

It therefore follows as a corollary that such a polynomial is -linear iff its roots form a -vector subspace of .