Additive polynomial

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

## Contents

* 1 Definition
* 2 Examples
* 3 The ring of additive polynomials
* 4 The fundamental theorem of additive polynomials
* 5 See also
* 6 References
* 7 External links

## Definition

Let k be a field of prime characteristic k. A polynomial P(x) with coefficients in k is called an additive polynomial, or a Frobenius polynomial, if

P(a+b)=P(a)+P(b)

as polynomials in a and b. 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.[1] For infinite fields the conditions are equivalent,[2] 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 q any multiple P of xq−x will satisfy P(a+b)=P(a)+P(b) for all a and b in the field, but will usually not be (absolutely) additive.

## Examples

The polynomial xp is additive.[1] Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem

(a+b)p=∑n=0p(pn)anbp−n.

Since p is prime, for all n=1,…,p−1 the binomial coefficient (pn) is divisible by p, which implies that

(a+b)p≡ap+bpmodp

as polynomials in a and b.[1]

Similarly all the polynomials of the form

τpn(x)=xpn

are additive, where n is a non-negative integer.[1]

## The ring of additive polynomials

It is quite easy to prove that any linear combination of polynomials τpn(x) with coefficients in k is also an additive polynomial.[1] An interesting question is whether there are other additive polynomials except these linear combinations. The answer is that these are the only ones.[3]

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[4]

k{τp}.

This ring is not commutative unless k is the field 𝔽p=ℤ/pℤ (see modular arithmetic).[1] Indeed, consider the additive polynomials ax and xp for a coefficient a in k. For them to commute under composition, we must have

(ax)p=axp,

and hence ap−a=0. This is false for a not a root of this equation, that is, for a outside 𝔽p.[1]

## The fundamental theorem of additive polynomials

Let P(x) be a polynomial with coefficients in k, and {w1,…,wm}⊂k be the set of its roots. 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 {w1,…,wm} forms a group with the field addition.[5]

## See also

* Drinfeld module
* Additive map

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Basic Structures of Function Field Arithmetic, Berlin: Springer, 1996, p. 1, doi:10.1007/978-3-642-61480-4, ISBN 3-540-61087-1
2. ↑ Goss 1996, p. 2, Proposition 1.1.5. sfn error: no target: CITEREFGoss1996 (help)
3. ↑ Goss 1996, p. 3, Corollary 1.1.6 harvnb error: no target: CITEREFGoss1996 (help)
4. ↑ Equivalently, Goss 1996, p. 1 harvnb error: no target: CITEREFGoss1996 (help) defines k{τp} to be the ring generated by τpn(x) and then proves (p. 3) that it consists of all additive polynomials.
5. ↑ Goss 1996, p. 4, Theorem 1.2.1. sfn error: no target: CITEREFGoss1996 (help)

## External links

* Weisstein, Eric W.. "Additive Polynomial". http://mathworld.wolfram.com/AdditivePolynomial.html.

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