On divisibility of class number of real abelian fields of prime conductor
For an odd prime \(p\), let \(\zeta_ p= e^{2\pi i/p}\) and denote by \(K\) the subfield of the real cyclotomic field \(\mathbb{Q}(\zeta_ p+ \zeta_ p^{-1})\) whose degree over \(\mathbb{Q}\) is a prime, say \(\ell\). This paper is a study of the divisibility of \(h_ K\), the class number of \(K\), by odd primes \(q<p\). Let \(C(K)\) denote the group of cyclotomic units of \(K\). The author first uses an arbitrary generator \(\gamma\) of \(C(K)\) to produce an element \(\rho\) of \(\mathbb{Z}[\zeta_ \ell]\) whose (absolute) norm is divisible by the primes \(q\) dividing \(h_ K\). By choosing \(\gamma\) as the ``standard'' generator of \(C(K)\), he is then able to write down an explicit expression for the corresponding \(\rho\). An inspection of that expression yields a series of smooth conditions for \(h_ K\not\equiv 0\pmod q\). Three examples: \(p=2\ell+1\) and \(\text{ord}_ \ell (q)= \ell- 1\); \(p= 4\ell+1\), \(q=3\) or 5 and \(\text{ord}_ \ell(q)= \ell-1\); \(p=2\ell+1\equiv 5\pmod 9\), \(q=3\) and \(\text{ord}_ \ell(3)= (\ell- 1)/2\). (Here \(\text{ord}_ \ell(q)\) denotes the exponent \(\text{mod }\ell\) of \(q\)).
scientific article; zbMATH DE number 492584