Twisted polynomial ring

In mathematics, a twisted polynomial is a polynomial over a field of characteristic

representing the Frobenius map

In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule for all

Over an infinite field, the twisted polynomial ring is isomorphic to the ring of additive polynomials, but where multiplication on the latter is given by composition rather than usual multiplication.

However, it is often easier to compute in the twisted polynomial ring — this can be applied especially in the theory of Drinfeld modules.

The twisted polynomial ring

is defined as the set of polynomials in the variable

It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation

Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials.

As an example we perform such a multiplication The morphism defines a ring homomorphism sending a twisted polynomial to an additive polynomial.

Here, multiplication on the right hand side is given by composition of polynomials.

we have the Freshman's dream

The homomorphism is clearly injective, but is surjective if and only if

The failure of surjectivity when

is finite is due to the existence of non-zero polynomials which induce the zero function on

[citation needed] Even though this ring is not commutative, it still possesses (left and right) division algorithms.