On cyclotomic units connected with \(p\)-adic characters
Let \(p\) be an odd prime and \(K\) an abelian number field containing any primitive \(p\)-th root \(\zeta\) of unity, of degree prime to \(p\). For an even character \(\chi\neq 1\) of the Galois group \(G\) of \(K\) over the rationals, we denote by \(\phi\),\({\bar\phi }\) the \(p\)-adic characters over \(\chi,\chi^{- 1}\omega\) respectively, where \(\omega\) is the character of \(G\) satisfying \(\zeta^{\omega (\sigma)}=\zeta^{\sigma}\) \((\sigma\in G)\). Here the values of any character are contained in an algebraic closure \(\Omega_p\) of the \(p\)-adic number field. For any abelian \(p\)-group \(S\) on which \(G\) acts we consider the \(\phi\)-part \(S_{\phi}\) and the \({\bar\phi}\)-part \(S_{{\bar\phi}}\) of \(S\). Let \(K_{\chi}\) be the fixed field of \(\mathrm{Ker }\chi\) and \(E_{\chi},C_{\chi}\) the unit group, the cyclotomic unit group of \(K_{\chi}\), respectively. A unit \(\eta_{\phi}\) in \(C_{\chi}\) is called a \(\phi\)-relative cyclotomic unit if \(\eta_{\phi} E^p_{\chi}\) generates \((C_{\chi} E^p_{\chi}/E^p_{\chi})_{\phi}\). Let \(m({\bar\phi}\)) be the exponent of the exact power of \(p\) dividing the Bernoulli number \(B_1(\chi\omega^{-1})\) in \(\Omega_p\). Further by \(A\) we mean the \(p\)-Sylow subgroup of the ideal class group of \(K\). One knows that the \(p^{m({\bar\phi})}\)-th power of any ideal representing a class in \(A_{{\bar\phi }}\) is principal. In the present paper it is proved that a \(\phi\)-relative cyclotomic mit \(\eta_{\phi}\) is a \(p\)-th power in \(K\) if and only if \(m({\bar\phi})>0\) and the \(p^{m({\bar\phi})}\)-th power of any ideal \({\mathfrak a}\), (\({\mathfrak a},p)=1\), representing a class in \(A_{{\bar\phi}}\), is generated by a \(p\)-primary integer in \(K\). Here an integer \(\alpha\) in \(K\) is said to be \(p\)-primary if \(K(\alpha^{1/p})/K\) is an unramified extension.
scientific article; zbMATH DE number 3871494