Twisted polynomial ring

In mathematics, a twisted polynomial is a polynomial over a field of characteristic p {displaystyle p} in the variable τ {displaystyle au } representing the Frobenius map x ↦ x p {displaystyle xmapsto x^{p}} . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule In mathematics, a twisted polynomial is a polynomial over a field of characteristic p {displaystyle p} in the variable τ {displaystyle au } representing the Frobenius map x ↦ x p {displaystyle xmapsto x^{p}} . In contrast to normal polynomials, multiplication of these polynomials is not commutative, but satisfies the commutation rule for all x {displaystyle x} in the base field. 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. Let k {displaystyle k} be a field of characteristic p {displaystyle p} . The twisted polynomial ring k { τ } {displaystyle k{ au }} is defined as the set of polynomials in the variable τ {displaystyle au } and coefficients in k {displaystyle k} . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation τ x = x p τ {displaystyle au x=x^{p} au } for x ∈ k {displaystyle xin k} . Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials.