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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 .

Polynomial

This entry contributed by José Gallardo Alberni

## References

Goss, D. Basic Structures of Function Field Arithmetic. Berlin: Springer-Verlag, pp. 1-33, 1996.