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Power roots of polynomials over arbitrary fields

Published online by Cambridge University Press:  17 April 2009

Vincenzo Acciaro
Vincenzo Acciaro
Affiliation:
School of Computer Science Carelton University Ottawa, Ontario K1S 5B6, Canada
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Abstract

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Let F be an arbitrary field, and f(x) a polynomial in one variable over F of degree ≥ 1. Given a polynomial g(x) ≠ 0 over F and an integer m > 1 we give necessary and sufficient conditions for the existence of a polynomial z(x) ∈ F[x] such that z(x)m ≡ g(x) (mod f(x)). We show how our results can be specialised to ℝ, ℂ and to finite fields. Since our proofs are constructive it is possible to translate them into an effective algorithm when F is a computable field (for example, a finite field or an algebraic number field).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 2 , October 1994 , pp. 327 - 335
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Berlekamp, E.R., ‘Factoring polynomials over large finite fields’, Math. Camp. 24 (1970), 713–735.CrossRefGoogle Scholar
[2]Koblitz, N., p-adic numbers, p-adic analysis and zeta functions, (second edition) (Springer-Verlag, Berlin, Heidelberg, New York, 1984).CrossRefGoogle Scholar
[3]Miller, J.B., ‘Power roots of polynomials’, Bull. Austral. Math. Soc. Vol. 47 (1993), 163–168.CrossRefGoogle Scholar
[4]Lang, S., Algebra, (third edition) (Addison-Wesley, Reading, Mass., 1993).Google Scholar
[5]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia of Math. and its Applications 20, (Addison-Wesley, Reading, Mass., 1983).Google Scholar
[6]Pierce, R.S., Associative algebras (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[7]Rabin, M.O., ‘Probabilistic algorithms in finite fields’, SIAM J. Comput. 9 (1980), 273–280CrossRefGoogle Scholar
[8]Williams, K.S. and Hardy, K., ‘A refinement of H.C. Williams' q–th root algorithm’, Math. Comput. 61 (1993), 475–483.Google Scholar