In mathematics, the additive polynomials are an important topic in classical algebraic number theory.

Definition

be a field of prime characteristic. A polynomial with coefficients is called an additive polynomial, or a Frobenius polynomial, if

, such as its algebraic closure.

Occasionally absolutely additive is used for the condition above, and additive in the field. For infinite fields the conditions are equivalent, but for finite fields in the field, but will usually not be (absolutely) additive.

Examples

is additive. one has by the binomial theorem

the binomial coefficient is divisible, which implies that

.

Similarly all the polynomials of the form

is a non-negative integer.

.

The ring of additive polynomials

It is quite easy to prove that any linear combination 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.

. These imply that the additive polynomials form a ring under polynomial addition and composition. This ring is denoted

This ring is not commutative (see modular arithmetic).. For them to commute under composition, we must have

not a root

The fundamental theorem of additive polynomials

is separable is additive if and only if forms a group with the field addition.

See also

• Drinfeld module

• Additive map

References

External links