### People interested

Xavier Caruso, Jérémy Le Borgne

### Description

If k is a field and \sigma a ring endomorphism of k, the ring of skew polynomials k[X,\sigma] is the usual vector space of polynomials over k equipped with the multiplication deduced from the rule X a = \sigma(a) X (a \in K)

This ring is closely related to \sigma-modules over k and, consequently, to Galois representations.

The aim of the project is to implement usual functions on k[X,\sigma] when k is a finite field.

### Progress

A class has been written (for now, in python). It supports the following functions:

- basic arithmeric (addition, multiplication, euclidean division, gcd)
computation of the center Z(k[X,\sigma]) -- need to add a coercion map

computation of the so-called map \Psi : k[X,\sigma] \to Z(k[X,\sigma])

computation of the associated Galois representation (via the corresponding \sigma-module)

- factorization -- in progress

### Bugs

Do not derive from PolynomialRing_general since this class assumes that the variable commutes with the constants (probably rather hard: need to rewrite many things)