In mathematics, a twisted polynomial is a polynomial over a field of characteristic
Contents
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.
Definition
Let
As an example we perform such a multiplication
Properties
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. For example
using the fact that in characteristic
The homomorphism is clearly injective, but is surjective if and only if
Even though this ring is not commutative, it still possesses (left and right) division algorithms.