Simplification of expressions involving radicals
These two closely related (and interactively written) papers deal with the problem of ''denesting'' (decreasing the depth of) a nested radical, e.g., \(\sqrt{(5+2\sqrt{6})}=\sqrt{2}+\sqrt{3}\), or \(\sqrt{(4+3\sqrt{2})}=^ 4\sqrt{2}+^ 4\sqrt{8}\), or \({}^ 4\sqrt{- 4}=1+\sqrt{-1}\), etc. There are many more provocative curiosities, e.g., see \textit{D. Shanks} [Fibonacci Q. 12, 271, 280 (1974; Zbl 0294.12001)], including combinations of radicals which denest collectively but not individually. This is a matter for symbolic computation since we must identify equivalent answers by machine. In principle, this requires discovering a (possibly larger) field in which to use a rational basis to treat radicals as polynomials and also a simplification process (because the most standard form might not be the most easily recognized). The complexity seems generally exponential. The first paper deals with the quadratic case. With restriction to reals only the fourth root can enter additionally. A program is given for denesting; an essential feature is the recognition of perfect squares (e.g., of the norms of the radicands). The unpleasant surprise is that the nest of depth n is resolved into two nests of depth n-1, thus increasing the depth when \(n>2.\) The second paper deals more theoretically with Kummer extensions, using illustrations of degree 2 and 3. As before, we must first discover the larger working field. The earliest reference is to \textit{A. Capelli} [Rend. Accad. Sci. Fis. Mat., Napoli, III. Ser. 3, 243-252 (1897))]. The almost simultaneous Hilbert Zahlbericht escapes notice, but not the Artin-Tate cohomology, which is used in an appendix to find the Galois group of a Kummer field. Work of \textit{M. Kneser} [Acta. Arith. 26, 307- 308 (1975; Zbl 0314.12001)] and \textit{F. Halter-Koch} [J. Algebra 63, 318- 330 (1980; Zbl 0428.12020)] is cited to show that non-Kummerian extensions (with insufficient roots of unity) may also be used.
scientific article; zbMATH DE number 3916397