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The reducibility theorem for linearised polynomials over finite fields

Published online by Cambridge University Press:  17 April 2009

Stephen D. Cohen
Affiliation:
Department of MathematicsUniversity of GlasgowGlasgow G12 8QWScotland
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Abstract

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A self-contained elementary account is given of the theorem of S. Agou that classifies all composite irreducible polynomials of the form over a finite field of characteristic p. Written to appeal to a wide readership, it is intended to complement the original rather technical proof and other contributions by the author and by Moreno.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Agou, S., ‘Irréducibilité des polynômes sur un corps fini Fps’, J. Reine Agnew. Math. 292 (1977), 191195.Google Scholar
[2]Agou, S., ‘Factorisation sur un corps fini Fpn des polynômes composés lorsque f(X) est un polynôme irréductible de ’, J. Number Theory 9 (1977), 229239.CrossRefGoogle Scholar
[3]Agou, S., ‘Irréductibilité des polynômes sur un corps fini ’, J. Number Theory 10 (1978), 6469. 11 (1979) 20.CrossRefGoogle Scholar
[4]Agou, S., ‘Irréductibilité des polynômes sur un corps fini ’, Caned. Math. Bull. 23 (1980), 207212.CrossRefGoogle Scholar
[5]Agou, S., ‘Sur la factorisation des polynômes sur un corps fini ’, J. Number Theory 12 (1980), 447459.Google Scholar
[6]Berlekamp, E.R., Algebraic Coding Theory (McGraw-Hill, New York, 1968).Google Scholar
[7]Cohen, S.D., ‘The irreducibility of compositions of linear polynomials over a finite field’, Compositio Math. 47 (1982), 149152.Google Scholar
[8]Lidl, R. and Niederreiter, H., ‘Finite Fields’, in Encyclopedia Math. App. Vol. 20, now distributed by Cambridge University Press (Addison Wesley, Readitig, Mass.).Google Scholar
[9]Long, A.F., ‘Factorisation of irreducible polynomials over a finite field with the substitution for x’, Acta Arith. 25 (1973), 6580.CrossRefGoogle Scholar
[10]Long, A.F. and Vaughan, T.P., ‘Factorisation of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear I, II’, Linear Algebra 11 (1975), 5372. 13 (1976), 207–221.CrossRefGoogle Scholar
[11]Moreno, O., ‘Discriminants and the irreducibility of a class of polynomials’, Lect. Notes in Comput. Sci. 228 (1988), 178181.Google Scholar
[12]Moreno, O., ‘Discriminants and the irreducibility of a class of polynomials in a finite field of arbitrary characteristic’, J. Number Theory 28 (1988), 6265.Google Scholar