From: Joe Keane Subject: radicals Date: Thu, 3 Aug 2000 02:29:31 -0700 Newsgroups: sci.math.research To: sci-math-research@moderators.isc.org Summary: What's good? Keywords: nest What's a good reference on simplification of radical expressions? Some rules for simple expressions are given in high-school algebra, and anything further than that seems to be not well known. In fact, this area seems to be beneath most mathematicians since i find silly things in published papers. What i know now is some trivia, some heuristics, and various identities that may improve things or just lead in circles. Ideally we'd have some sort of reduction to canonical form, so we can see right away if two expressions are equal and so on. I suppose minimal polynomials are something like this, but they're not satisfying. -- Joe Keane, amateur mathematician ============================================================================== From: Dave Rusin Subject: Re: radicals Date: Thu, 3 Aug 2000 14:50:28 -0500 (CDT) To: jgk@jgk.org Joe, this issue came up last year on sci.math -- see 99/radicals Susan Landau has had a couple of papers on this topic; Math Reviews articles are attached. Perhaps you should look at some of these first, and then if you are still not satisfied your query to sci.math.research can be more pointed? dave (moderator) 95d:11144 11R06 Landau, Susan(1-MA-C) How to tangle with a nested radical. Math. Intelligencer 16 (1994), no. 2, 49--55. Starting from equations such as $$\root 3\of {\root 3\of 2-1}=\root 3\of {\tfrac 19}-\root 3\of {\tfrac 29}+\root 3\of {\tfrac 49}$$ the paper gives a very readable survey of recent results concerning reduction ("denesting") as above of composite radical expressions. Reviewed by Chr. U. Jensen _________________________________________________________________ 92k:12008 12Y05 (11Y16 68Q40) Landau, Susan(1-MA-C) Simplification of nested radicals. SIAM J. Comput. 21 (1992), no. 1, 85--110. 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 necessary and sufficient conditions for a denesting are given, and the algorithm computes a denesting of $\alpha$ when it exists. If the base field does not contain all roots of unity, then it is shown how to compute a denesting that is within one of optimal over the base field adjoining a primitive $l$-root of unity. The algorithms require computing the splitting field of the minimal polynomial of $\alpha$ over $k$, and have exponential running time. The proofs need a clever use of Galois theory, Kummer theory and Hilbert's Theorem 90. Reviewed by Maurice Mignotte © Copyright American Mathematical Society 2000