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On the degree of repeated radical extensions. (English) Zbl 07451291

L. J. Mordell [Pac. J. Math. 3, 625–630 (1953; Zbl 0051.26801)] proposed the following problem: let \(K\) be an algebraic number field and \(x_1, x_2, \ldots, x_s\), be algebraic numbers of degrees \(n_1, n_2, \ldots, n_s\), over \(K\), respectively. When does the field \(K(x_1, x_2,\ldots, x_s)\) have degree \(n_1n_2\ldots n_s\) over \(K\)?
Many authors have studied this problem in the case of repeated radical extensions \(K(x_1, x_2,\ldots, x_s)\) of \(K\) (this means that for each \(i\), \(1\leq i \leq s\), some power \({x_i}^{n_i}\) of \(x_i\) lies in \(K\)). More precisely, the special case of radical extension \(K(x_1)\) was solved by K. Th. Vahlen [Acta Math. 19, 195–198 (1895; JFM 26.0121.02)] if \(K = Q\), A. Capelli [Napoli Rend. (3) 3, 243–252 (1897; JFM 28.0090.01)] if \(K\) is an arbitrary field of characteristic \(0\), and [L. Redei, Algebra. Vol. 1. Oxford: Pergamon Press (1967; Zbl 0191.00502)] in general. Under the Kummerian hypothesis, the case \(s> 1\) has been studied by H. Hasse [Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. 2. Aufl. Würzburg etc.: Physica-Verlag (1965; Zbl 0138.03202)] and without this hypothesis by A. S. Besicovitch [J. Lond. Math. Soc. 15, 3–6 (1940; Zbl 0026.20301)]. Their results are contained in the following theorem of Mordell. If \(K\) contains all the \(n_1\)-th, \(n_2\)-th,…, \(n_s\)-th roots of unity, or that \(K\) is a subfield of \(\mathbb{R}\) with all \(x_1,\ldots, x_s\) are real, then the extension \(K(x_1, x_2, x_s) \) has degree \(n_1\ldots n_s\) over \(K\) if and only if there exists no relation of the form \({x_1}^{\varepsilon_1}\ldots {x_s}^{\varepsilon_s}=a\), where \(a\) is a number in \(K\), unless \(\varepsilon_1 = 0\pmod{n_1}, \varepsilon_2 = 0\pmod{n_2}, \ldots, \varepsilon_s = 0\pmod{n_s}\).
The aim of this paper (under review) is to give necessary and sufficient conditions for \[ [F(\sqrt[n_1]{M_1},\ldots, \sqrt[n_s]{M_s}): F] = n_1 \ldots n_s, \] where \(F\) is an arbitrary field of characteristic not dividing any \(n_i\) and \(M_1,\ldots, M_s\in F\). In particular, the above problem posed by Mordell in the case of repeated radical extensions has been solved. By virtue of Isaacs’ result [I. M. Isaacs, Proc. Am. Math. Soc. 25, 638–641 (1970; Zbl 0203.34902)], the author (of the paper under review) also pointed out that the relation \[ F[\sqrt[n_1]{M_1},\ldots, \sqrt[n_s]{M_s}]=F[b_1\sqrt[n_1]{M_1}+\cdots + b_s\sqrt[n_s]{M_s}] \] holds, under certain conditions, for any nonzero \(b_1, \ldots, b_s\in F\).

MSC:

12F05 Algebraic field extensions
12F10 Separable extensions, Galois theory
12E05 Polynomials in general fields (irreducibility, etc.)
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References:

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