A modular GCD algorithm over number fields presented with multiple extensions

Authors:
Mark van Hoeij;Michael Monagan
Affiliations:
Florida State University, Tallahassee, FL;Simon Fraser University, Burnaby, B.C. Canada
Venue:
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Year:
2002

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Abstract

We consider the problem of computing the monic gcd of two polynomials over a number field L = ℚ(α1,…,αn). Encarnacion, Langemyr and McCallum have already shown how Brown's modular GCD algorithm for polynomials over ℚ can be modified to work for ℚ(α).Our first contribution is an extension of Encarnacion's modular GCD algorithm to the case n 1 without converting to a single field extension. Our second contribution is a proof that it is not necessary to test if p divides the discriminant. This simplifies the algorithm; it is correct without this test.Our third contribution is the design of a data structure for representing multivariate polynomials over number fields with multiple field extensions. We have a complete implementation of the modular GCD algorithm using it. We provide details of some practical improvements.