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In this paper we consider in detail the composition of an irreduciblepolynomial with X^2 and suggest a recurrent construction of irreduciblepolynomials of fixed degree over finite fields of odd characteristics. Moreprecisely, given an irreducible polynomial of degree n and order 2^rt with todd, the construction produces ord_t(2)/d irreducible polynomials of degree nand order t for a certain divisor d of n. The construction can be used, forexample, to search irreducible polynomials with specific requirements on itscoefficients.

A recurrent construction of irreducible polynomials of fixed degree over finite fields