In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
It is equivalent to assume that this equality holds for all a and b in some infinite field containing k, such as its algebraic closure.
Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that P(a + b) = P(a) + P(b) for all a and b in the field.
For infinite fields the conditions are equivalent, but for finite fields they are not, and the weaker condition is the "wrong" as it does not behave well.
Similarly all the polynomials of the form are additive, where n is a non-negative integer.
The definition makes sense even if k is a field of characteristic zero, but in this case the only additive polynomials are those of the form ax for some a in k.[citation needed] It is quite easy to prove that any linear combination of polynomials
An interesting question is whether there are other additive polynomials except these linear combinations.
These imply that the additive polynomials form a ring under polynomial addition and composition.
Indeed, consider the additive polynomials ax and xp for a coefficient a in k. For them to commute under composition, we must have and hence ap − a = 0.
Assuming that the roots of P(x) are distinct (that is, P(x) is separable), then P(x) is additive if and only if the set
forms a group with the field addition.