## 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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