Semi-characteristic polynomials, ϕ-modules and skew polynomials

Jérémy Le Borgne (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We introduce the notion of semi-characteristic polynomial for a semi-linear map of a finite- dimensional vector space over a field of characteristic p. This polynomial has some properties in common with the classical characteristic polynomial of a linear map. We use this notion to study skew polynomials and linearized polynomials over a finite field, giving an algorithm to compute the splitting field of a linearized polynomial over a finite field and the Galois action on this field. We also give a way to compute the optimal bound of a skew polynomial. We then look at properties of the factorizations of skew polynomials, giving a map that computes several invariants of these factorizations. We also explain how to count the number of factorizations and how to find them all.

Mots clés: skew polynomials; phi-modules; finite fields; galois representations

Domaines: Théorie des représentations [math.RT]

Jérémy Le Borgne. Semi-characteristic polynomials, ϕ-modules and skew polynomials. 2011. ⟨hal-00594597⟩