On the normalization of numbers and functions defined by radicals
The problem addressed in the paper under review is the efficient computation of a basis for the field extension \(K/k\) where \(K=k(A)\), \(k\) is a field of characteristic \(0\), \(A= \{\root n \of {\beta _1}, \ldots, \root n \of {\beta _s}\}\), \(\beta _1,\ldots, \beta _s \in k\) and \(n\) is a positive integer. The author presents a method based on Kummer theory and \textit{M. Kneser}'s theorem [Acta Arith. 26, 307-308 (1975; Zbl 0314.12001)]. In fact \textit{R. Zippel} [J. Symb. Comput. 1, 189-210 (1985; Zbl 0574.12002)] gave an algorithm adding \(n\)th roots of unity to the ground field. This addition increases time computation. Using Kneser's work the author avoids unnecessary extensions by roots of unity. Section 2 presents the theoretical results needed in the paper as well as the main mathematical results. In Section 3 algorithmic methods are given avoiding as much as possible factorization over algebraic extensions of \(k\) and the introduction of spurious roots of unity. The approach is based on Zippel's work but includes several improvements using the results of Section 2 as well as a different presentation based on Hermite and Smith forms preferred for programming purposes.
scientific article; zbMATH DE number 1934783
scientific article