Twisted polynomial ring explained
in the variable
representing the
Frobenius map
. In contrast to normal polynomials, multiplication of these polynomials is not
commutative, but satisfies the commutation rule
for all
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.
Definition
Let
be a field of characteristic
. The twisted polynomial ring
is defined as the set of polynomials in the variable
and coefficients in
. It is endowed with a
ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation
for
. Repeated application of this relation yields a formula for the multiplication of any two twisted polynomials.
As an example we perform such a multiplication
(a+b\tau)(c+d\tau)=a(c+d\tau)+b\tau(c+d\tau)=ac+ad\tau+bcp\tau+bdp\tau2
Properties
The morphism
k\{\tau\}\tok[x], a0+a1\tau+ … +a
a0x+a
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
(ax+bxp)\circ(cx+dxp)=a(cx+dxp)+b(cx+dxp)p=acx+adxp+bcpxp+bdpx
,
using the fact that in characteristic
we have the
Freshman's dream
.
The homomorphism is clearly injective, but is surjective if and only if
is infinite. The failure of surjectivity when
is finite is due to the existence of non-zero polynomials which induce the zero function on
(e.g.
over the finite field with
elements).
Even though this ring is not commutative, it still possesses (left and right) division algorithms.