# Algorithms for a multiple algebraic extension II

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## Abstract

We give a fast algorithm for computing the greatest common divisor of two univariate polynomials over a multiple algebraic extension of the rational numbers. The algorithm is almost linear in terms of the output length, *i.e.*, it works in time *O*(*d*^{1+δ}, for all δ>0, where *d* is an a priori bound on the length of the output. Since we require time Ω(*d*) just to write down the output the algorithm is close to optimal. The algorithm uses a technique referred to as dynamic evaluation for computing in algebraic extensions defined by reducible polynomials.

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© Springer-Verlag Berlin Heidelberg 1991