The Ring of Additive Polynomials
It is quite easy to prove that any linear combination of polynomials with coefficients in k is also an additive polynomial. An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones.
One can check that if P(x) and M(x) are additive polynomials, then so are P(x) + M(x) and P(M(x)). These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted
This ring is not commutative unless k equals the field (see modular arithmetic). Indeed, consider the additive polynomials ax and xp for a coefficient a in k. For them to commute under composition, we must have
or ap − a = 0. This is false for a not a root of this equation, that is, for a outside
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