On the Orders of the Elements of a Square Extension of a Finite Field of Characteristic 2

Abstract

Let F(2m) will be an arbitrary finite field of characteristic 2. It’s square extension will be considered as an algebra with basiс elements 1 and e over the field F(2m). Here 1 is considered as the unit element of the algebra, and e satisfies the relation: e2 = e + α. An element α maybe arbitrary from the field F(2m), but it is not satisfying to the condition α = x + x2 for some x element from F(2m). Let us n0 (α) denote the order of basis element e. Then the main result of the paper can be formulated as: The irreducible polynomial 1 + t + αt2 divides the polynomial 1 + tn if and only the order if n0 (α) divides a natural n. The similar results for arbitrary elements of field F(2m) follow from main theorem. The proof of main result based on the properties of the recurrence relations between the polynomials Pn (α) and Qn (α), definite for all n = 0, 1, 2, … by the relations en = Pn (α) + Qn (α) e. The formulas for the generating series of these polynomials contain the most important such properties. The formulas were obtained.

Keywords

• finite field
• characteristic 2
• order of element
• square extension over a field
• irreducible polynomial
• recursive sequence
• generating series