CM-fields with cyclic ideal class groups of 2-power orders
Using [\textit{J. Hoffstein}, Invent. Math. 55, 37-47 (1979; Zbl 0474.12009)], we prove that there are only finitely many CM-fields \({\mathcal N}\) with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of \({\mathcal N}\). Since their actual determination would be too difficult, we only content ourselves with the determination of the non-quadratic imaginary cyclic number fields of 2-power degrees with cyclic ideal class groups of 2-power orders. There are exactly 22 much number fields. This present determination is a nice complement to the determination of all non-quadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents \(\leq 2\) completed in [\textit{S. Louboutin}, Math. Comput. 64, 323-340 (1995; Zbl 0822.11072)]. This determination stems from the following good lower bound on the relative class numbers \(h_{\mathcal N}^-\) of the non-quadratic imaginary cyclic number fields of 2-power degrees \(2n= 2^r\geq 4\) and conductors \(f_{\mathcal N}\): \[ h_{\mathcal N}^-\geq \frac{2\varepsilon_{\mathcal N}} {e(2n-1)} \Biggl( \frac{\sqrt{f_{\mathcal N}}} {\pi(\log f_{\mathcal N}+0.05)} \Biggr)^n, \] where \(\varepsilon_{\mathcal N}\) is asymptotic to 1 when \(f_{\mathcal N}\) goes to infinity.
cyclic ideal class groups of 2-power orders
scientific article; zbMATH DE number 1097545
scientific article