On the class number of an abelian field with prime conductor (Q1318933)

The author intends to generalize \textit{K. Uchida}'s result on the class number of a cubic cyclic field [J. Number Theory 8, 346-349 (1976; Zbl 0341.12006)]. Namely, let \(\ell\) and \(p\) be two odd primes satisfying \(p \equiv 1 \pmod \ell\) and \(2^ a\) be the maximal 2-power factor of \((p- 1)/ \ell\). Moreover, let \(K\) be the imaginary subfield of the \(p\)-th cyclotomic field \(\mathbb{Q} (\zeta_ p)\) of degree \(2^ a \ell\), \(K_ 0\) be the maximal real subfield of \(K\) and \(L\) be the subfield of \(K_ 0\) of degree \(\ell\). Under these circumstances, he proves that if 2 is a primitive root modulo \(\ell\), then the conditions (i) the relative class number of \(K\) is even, (ii) the class number of \(K_ 0\) is even, (iii) the class number of \(L\) is even, are all equivalent. Moreover, he provides a necessary and sufficient condition for these in terms of 2-rank of the subgroup of totally positive units in the group generated by the cyclotomic units in \(L\).