Additive polynomial

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

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

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. For infinite fields the conditions are equivalent, but for finite fields they are not, and the weaker condition is the "wrong" one and does not behave well. For example, over a field of order q any multiple P of x^q − 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.

The polynomial x^p is additive. Indeed, for any a and b in the algebraic closure of k one has by the binomial theorem

Since p is prime, for all n = 1, ..., p−1 the binomial coefficient ${\displaystyle \scriptstyle {p \choose n}}$ is divisible by p, which implies that

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