A generalization of Swan’s theorem
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- by Harold M. Fredricksen, Alfred W. Hales and Melvin M. Sweet PDF
- Math. Comp. 46 (1986), 321-331 Request permission
Abstract:
Let f and g denote polynomials over the two-element field. In this paper we show that the parity of the number of irreducible factors of ${x^n}f + g$ is a periodic function of n, with period dividing eight times the period of the polynomial ${f^2}(x(g/f)’- n(g/f))$. This can be considered a generalization of Swan’s trinomial theorem [3].References
- Elwyn R. Berlekamp, Algebraic coding theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0238597
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- Richard G. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12 (1962), 1099–1106. MR 144891, DOI 10.2140/pjm.1962.12.1099
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Math. Comp. 46 (1986), 321-331
- MSC: Primary 11T06
- DOI: https://doi.org/10.1090/S0025-5718-1986-0815852-1
- MathSciNet review: 815852