A note on class number 1 criteria for totally real algebraic number fields (Q2773292)

\textit{M. Kutsuna} applied Rabinowitsch's method for imaginary quadratic fields to real quadratic fields and obtained some class number one criteria [cf. Nagoya Math. J. 79, 123-129 (1980; Zbl 0447.12006)]. In order to extend Kutsuna's result to arbitrary totally real algebraic number fields, in this paper the author uses Siegel's formula for the special values of Dedekind zeta-functions attached to them. NEWLINENEWLINENEWLINENamely, let \(K\) be a totally real algebraic number field of degree \(n\) and \(\delta\) be the different of \(K\). Moreover, for a natural number \(\ell\) let \(T_\ell\) be the set of all totally positive elements of \(K\) in \(\delta^{-1}\) with given trace \(\ell\). Set \(T= \bigcup_{\ell=1}^r T_\ell\), where \(r\) is the dimension over \(\mathbb{C}\) of the space of modular forms of weight \(2n\). Then he proves first the following: the class number of \(K\) is equal to 1 if and only if the ideal \((\nu)\delta\) can be written as a product of powers of principal prime ideals of \(K\) for all \(\nu\) in \(T\). From this, he obtains the following class number 1 criterion similar to Rabinowitsch-Kutsuna: Let \(K\) be a totally real algebraic number field of degree \(n>1\), whose different \(\delta\) is \((\beta)\) for some \(\beta\) in \(K\). Then, if \(\mathbb{N}_{K/\mathbb{Q}} (\nu\beta)\) is \(\pm 1\) or a prime for any \(\nu\) in \(T\), then the class number of \(K\) is equal to 1.