Linearized polynomials over finite fields revisited

We give new characterizations of the algebra Ln(Fqn) formed by all linearized polynomials reduced modulo (xqn−x) over the finite field Fqn after briefly surveying some known ones. One isomorphism we construct is between Ln(Fqn) and the composition algebra Fqn∨⊗FqFqn. The other isomorphism we construct is between Ln(Fqn) and the so-called Dickson matrix algebra Dn(Fqn). We also further study the relations between a linearized polynomial and its associate Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us new approaches to establish some special forms of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra Ln(Fqm) which is formed by all linearized polynomials reduced modulo (xqn−x) over a subfield Fqm of Fqn where m|n is also described.