  In this paper, we study the arithmetics of skew polynomial rings over finite
fields, mostly from an algorithmic point of view. We give various algorithms
for fast multiplication, division and extended Euclidean division. We give a
precise description of quotients of skew polynomial rings by a left principal
ideal, using results relating skew polynomial rings to Azumaya algebras. We use
this description to give a new factorization algorithm for skew polynomials,
and to give other algorithms related to factorizations of skew polynomials,
like counting the number of factorizations as a product of irreducibles.

Number Theory