Abstract
AbstractLet D be a unital associative division ring and D[t, σ, δ] be a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation. For each f ϵ D[t, σ, δ] of degree m > 1 with a unit as leading coefficient, there exists a unital nonassociative algebra whose behaviour reflects the properties of f. These algebras yield canonical examples of right division algebras when f is irreducible. The structure of their right nucleus depends on the choice of f. In the classical literature, this nucleus appears as the eigenspace of f and is used to investigate the irreducible factors of f. We give necessary and sufficient criteria for skew polynomials of low degree to be irreducible. These yield examples of new division algebras Sf.
Highlights
The investigation of skew polynomials is an active area in algebra which has applications to coding theory, to solving differential and difference equations, and in engineering, to name just a few
Let D be a unital associative division ring and R = D[t; σ, δ] a skew polynomial ring, where σ is an endomorphism of D and δ a left σ-derivation
The first one considers the structure of the right nucleus of the algebras Sf, establishing how it reflects the type of the skew polynomial f it is defined with, and the important role irreducible polynomials play in the construction of classes of nonassociative unital division algebras
Summary
The investigation of skew polynomials is an active area in algebra which has applications to coding theory, to solving differential and difference equations, and in engineering, to name just a few. The first one considers the structure of the right nucleus of the algebras Sf , establishing how it reflects the type of the skew polynomial f it is defined with, and the important role irreducible polynomials play in the construction of classes of nonassociative unital (right) division algebras. The second part looks at skew polynomials of low degree as well as the polynomial f (t) = tm − a, and when these polynomials are irreducible in D[t; σ, δ], in order to obtain examples for the construction of (right) division algebras. We collect some irreducibility criteria for polynomials of low degree and the polynomial f (t) = tm − a in both R = D[t; σ] and R = D[t; σ, δ] in Sections 5 and 6, including the special case where D is a finite field. Most of this work is part of the first author’s PhD thesis [4] written under the supervision of the second author
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