On the Stickelberger ideal of (2,\dots, 2)-extensions of a cyclotomic number field
For rational integers \(m_1,\ldots,m_t\), let \(K\) denote the abelian field generated by \(\sqrt{m_1},\ldots,\sqrt{m_t}\) over the cyclotomic field of \(p^n\)-th roots of 1 \((p\) an odd prime, \(n\ge 1)\). The author constructs a system of generators for the minus part of the Stickelberger ideal \(S\) of \(\mathbb Z[G]\), where \(G=\mathrm{Gal}(K/\mathbb Q)\). This gives an expression for the index \((\mathbb Z[G]^-:S^-)\) in terms of certain determinants \(D_{p^n}\) with elements in \(\mathbb Z\). Upon calculating these determinants (each of which depends on a character \(\chi\) of \(K)\) one arrives at a formula connecting this index to the relative class number of \(K\). The formula in fact is a special case of one proved by \textit{C.-G. Schmidt} [J. Number Theory 11, No. 1, 128--159 (1979; Zbl 0403.12009)] for arbitrary abelian fields. Finally, the author finds a natural relationship between the \(p\)-divisibility of \(D_p\) and the \(\chi\)- irregularity of \(p\).
scientific article; zbMATH DE number 4177202