Separable Automorphisms on Matrix Algebras over Finite Field Extensions

Abstract

Let (F ⊆ K) an extension of finite fields and (A = Mn K) be the ring of square matrices of order n over (K) viewed as an algebra over (F). Given an (F)--automorphism (σ) on (A) the Ore extension (A[z;σ]) may be used to built certain convolutional codes, namely, the ideal codes. We provide an algorithm to decide if the automorphism (σ) on (A) is a separable returning the corresponding separability element (p). In this case (p) is also a separability element for the extension (F[z] ⊆ A[z;σ]), and as a consequence ideal codes are generated by idempotents in (A[z;σ]), which can be computed applying previous algorithms of the authors.