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Linearized polynomials over finite fields revisited. (English) Zbl 1345.11084
Summary: We give new characterizations of the algebra \(\mathcal L_n(\mathbb F_{q^n})\) formed by all linearized polynomials reduced modulo \((x^{q^n}-x)\) over the finite field \(\mathbb F_{q^n}\) after briefly surveying some known ones. One isomorphism we construct is between \(\mathcal L_n(\mathbb F_{q^n})\) and the composition algebra \(\mathbb F_{q^n}^{\vee}\otimes_{\mathbb F_q}\mathbb F_{q^n}\). The other isomorphism we construct is between \(\mathcal L_n(\mathbb F_{q^n})\) and the so-called Dickson matrix algebra \(\mathcal D_n(\mathbb F_{q^n})\). 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. The 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 \(\mathcal L_n(\mathbb F_{q^m})\) which is formed by all linearized polynomials reduced modulo \((x^{q^n}-x)\) over a subfield \(\mathbb F_{q^m}\) of \(\mathbb F_{q^n}\) where \(m\mid n\) is also described.

11T06 Polynomials over finite fields
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
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