@MISC{Landau93simplificationof, author = {Susan Landau}, title = {Simplification of Nested Radicals}, year = {1993} }

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Abstract

Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then we give necessary and sufficient conditions for a denesting, and our algorithm computes a denesting of ff when it exists. If the base field does not contain all roots of unity, then we show how to compute a denesting that is within depth one of optimal over the base field adjoin a single root of unity. Throughout our paper, we choose to represent a primitive l th root of unity by its symbol i l , rather than as a nested radical. The algorithms require computing the splitting field of the minimal polynomial of ff over k, and have exponential running time. In his magic way, Ramanujan observed a number of striking relationships between certain nested radicals: 3 q 3 p 2 \Gamma 1 = 3 q 1=9 \Gamma 3 q 2=9 + 3 q 4=9 (1) q 3 p 5 \Gamma 3 p 4 = 1=3( 3 p 2 + 3 p 20 ...