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Irreducibility Criteria for Polynomials with non-negative Coefficients

Published online by Cambridge University Press:  20 November 2018

Michael Filaseta
Michael Filaseta*
Affiliation:
University of South Carolina, Columbia, South Carolina
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In [7, b.2, VIII, 128] Pólya and Szegö state the following theorem of A. Cohn:

THEOREM 1. Let dndn−x … d0 be the decimal representation of a prime. Then

is irreducible.

Thus, for example, since 1289 is prime, x3 + 2x2 + 8x + 9 is irreducible. Brillhart, Odlyzko, and the author generalized Cohn's Theorem in three different directions. As examples of these types of generalizations, we note the following results, the first two of which are special cases of a result in [1] and the third of a result in [3].

THEOREM 2. Let dndn−x … d0 be the base b representation of a prime where b is an integer ≧2. Then

is irreducible.

THEOREM 3. Let

be such that f(10) is prime and 0 ≦ dj ≦ 167 for j = 0, 1, …, n. Then f(x) is irreducible.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 339 - 351
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Brillhart, J., Filaseta, M. and Odlyzko, A., On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981), 1055–1059.Google Scholar
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