On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields. (English) Zbl 1130.11068

The authors consider the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field. They characterize these by employing the Stickelberger-Swan Theorem. In the case of binary fields, they present the conditions in terms of the exponents of the monomials of the corresponding self-reciprocal polynomials.


11T06 Polynomials over finite fields
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[1] Ahmadi, O., Self-reciprocal irreducible pentanomials over \(F_2\), Des. Codes Cryptogr., 38, 395-397 (2006) · Zbl 1172.11309
[2] Bluher, A., A Swan-like theorem, Finite Fields Appl., 12, 128-138 (2006) · Zbl 1105.11040
[3] Carlitz, L., Some theorems on irreducible reciprocal polynomials over a finite field, J. Reine Angew. Math., 227, 212-220 (1967) · Zbl 0155.09801
[4] Cohen, S. D., On irreducible polynomials of certain types in finite fields, Proc. Cambridge Philos. Soc., 66, 335-344 (1969) · Zbl 0177.06601
[5] Hales, A.; Newhart, D., Irreducibles of tetranomial type, (Mathematical Properties of Sequences and Other Combinatorial Structures (2003), Kluwer) · Zbl 1059.11071
[6] Lidl, R.; Niederreiter, H., Introduction to Finite Fields and Their Applications (1994), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0820.11072
[7] Meyn, H., On the construction of irreducible self-reciprocal polynomials over finite fields, Appl. Algebra Engrg. Comm. Comput., 1, 43-53 (1990) · Zbl 0724.11062
[9] Swan, R., Factorization of polynomials over finite fields, Pacific J. Math., 12, 1099-1106 (1962) · Zbl 0113.01701
[10] Varshamov, R. R.; Garakov, G. A., On the theory of selfdual polynomials over a Galois field, Bull. Math. Soc. Sci. Math. Roumaine (N.S.), 13, 403-415 (1969) · Zbl 0228.12003
[11] Yucas, J. L.; Mullen, G. L., Self-reciprocal irreducible polynomials over finite fields, Des. Codes Cryptogr., 33, 275-281 (2004) · Zbl 1146.11336
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