Semi-characteristic polynomials, φ-modules and skew polynomials

Details Semi-characteristic polynomials, φ-modules and skew polynomials Semi-characteristic polynomials, φ-modules and skew polynomials

2011-05-20

arxiv.org/abs/1105.4083v1

Mathematics

Representation Theory

Scientific paper

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.

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