Euclidean ideal classes in Galois number fields of odd prime degree (Q2164875)

It has been shown by \textit{H. W. Lenstra jun.} [Astérisque 61, 121--131 (1979; Zbl 0401.12005)] that only algebraic number fields with cyclic class-group \(Cl(K)\) can have a Euclidean ideal, and the General Riemann Hypothesis implies that every field \(K\) of unit rank \(r(K)\ge1\) and cyclic \(Cl(K)\) has an Euclidean ideal. Later \textit{J. M. Deshouillers} et al. [Math. Z. 296, No. 1--2, 847--859 (2020; Zbl 1485.11154)] showed that if \(r(K)\ge 3\), \(Cl(K)\) is cyclic, the Hilbert class-field \(H(K)\) of \(K\) is abelian over \(\mathbb Q\) and the extension \(\mathbb Q(\zeta_f)/K\) is cyclic (with \(f\) being the conductor of \(K\)), then \(K\) has a Euclidean ideal class. The authors prove that if \(K_1,K_2\) are Galois fields of odd prime degree and cyclic \(Cl(K)\), the fields \(H(K_1),H(K_2)\) are abelian over \(\mathbb Q\), and the extensions \(K_1K_2/K_i\) are ramified for \(i=1,2\), then at least one of the fields \(K_i\) has a Euclidean ideal class. This generalizes a result of the second author and \textit{S. Gun} [Mich. Math. J. 69, No. 2, 429--448 (2020; Zbl 1445.13018)] who proved this for cubic fields under certain stronger assumptions.