In mathematics, the additive polynomials are an important topic in classical algebraic number theory.
Definition
Let <math>k</math> be a field of prime characteristic <math>k</math>. A polynomial <math>P(x)</math> with coefficients in <math>k</math> is called an additive polynomial, or a Frobenius polynomial, if
<math display=block>P(a+b)=P(a)+P(b)</math>
as polynomials in <math>a</math> and <math>b</math>. It is equivalent to assume that this equality holds for all <math>a</math> and <math>b</math> in some infinite field containing <math>k</math>, such as its algebraic closure.
Occasionally absolutely additive is used for the condition above, and additive is used for the weaker condition that <math>P(a+b)=P(a)+P(b)</math> for all <math>a</math> and <math>b</math> 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. For example, over a field of order <math>q</math> any multiple <math>P</math> of <math>x^q-x</math> will satisfy <math>P(a+b)=P(a)+P(b)</math> for all <math>a</math> and <math>b</math> in the field, but will usually not be (absolutely) additive.
Examples
The polynomial <math>x^p</math> is additive.
One can check that if <math>P(x)</math> and <math>M(x)</math> are additive polynomials, then so are <math>P(x)+M(x)</math> and <math>P(M(x))</math>. These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted
<math display=block>k\{ \tau_p\}.</math>
This ring is not commutative unless <math>k</math> is the field <math>\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}</math> (see modular arithmetic).