On existence of Euclidean ideal classes in real cubic and quadratic fields with cyclic class group (Q2195017)

Let \(K_i\) be a number field with Hilbert class field \(H(K_i)\) which is abelian over \(\mathbb{Q}\) for \(i=1,2,3\). Let \(f_i\) be the conductor of \(H(K_i)\), that is, \(\mathbb{Q}(\zeta_{f_i})\) be the smallest cyclotomic field containing \(H(K_i)\) where \(\zeta_{f_i}\) is a primitive \(f_i^{th}\) root of unity for \(i=1,2,3\). Let \(f=\mathrm{l.c.m}(16,f_1,f_2,f_3)\) if \(K_i\) is real quadratic for \(i=1,2,3\) and \(f=l.c.m(16,f_1,f_2)\) if \(K_i\) is real cubic for \(i=1,2\). Let \(F=\mathbb{Q}(\zeta_f)\) where \(\zeta_f\) is the primitive \(f^{th}\) root of unity. Let \(\mathrm{Gal}(F/H(K_i))\) be the Galois group of \(H(K_i)\) for i=1,2,3 and \(G_l\) be the Galois group of \(F\) over \(\mathbb{Q}(\zeta_l)\) where either \(l\) is an odd prime dividing \(f\) or \(l=4\). Then the authors prove the following in this set up. \begin{enumerate} \item For two distinct real cubic fields \(K_i,i=1,2\) with prime class numbers and \(G\) the Galois group of \(F\) over \(K_1K_2\) and the other terms as defined above, if \(G\nsubseteq \bigcup_{l=4 \text{ or }l\mid f,\ l \text{ an odd prime}}G_l \cup \mathrm{Gal}(F/H(K_1)) \cup \mathrm{Gal}(F/H(K_2))\) then at least one of \(K_1,K_2\) has a Euclidean ideal Class. \item For two distinct real quadratic fields \(K_i,i=1,2,3\) with prime class numbers and \(G\) the Galois group of \(F\) over \(K_1K_2K_3\) and the other terms as defined above, if \(G\nsubseteq \bigcup_{l=4 \text{ or }l\mid f,\ l \text{ an odd prime}}G_l \cup \mathrm{Gal}(F/H(K_1)) \cup \mathrm{Gal}(F/H(K_2))\cup \mathrm{Gal}(F/H(K_3))\) then at least one of \(K_1,K_2,K_3\) has a Euclidean ideal Class. \item For two distinct real quadratic fields \(K_i,i=1,2\) with prime class numbers and \(G\) the Galois group of \(F\) over \(K_1K_2\) and \(f=l.c.m(16,f_1,f_2)\) with the other terms as defined above, if \(G\nsubseteq \bigcup_{l=4 \text{ or }l\mid f,\ l \text{ an odd prime}}G_l \cup \mathrm{Gal}(F/H(K_1)) \cup \mathrm{Gal}(F/H(K_2))\) then at least one of \(K_1,K_2\) has a Euclidean ideal Class provided the Elliot-Halberstram conjecture holds. \end{enumerate} The authors at the end, give explicit examples where these results can be applied.