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← Revision 8 as of 2012-02-23 19:44:31 ⇥
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| 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 $a X = \sigma(a) X$ ($a \in K$) | 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 $X a = \sigma(a) X$ ($a \in K$) |
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)