On Iwasawa \(\lambda_p\)-invariants of relative real cyclic extensions of degree \(p\) (Q1382856)

Let \(K\) be a totally real number field whose class number is not divisible by \(p\), and assume that \(L/K\) is a real cyclic extension of prime degree \(p\). Let \(K_\infty\) denote the cyclotomic \(\mathbb{Z}_p\)-extension of \(K\) and assume that there is only one prime in \(K_\infty\) above \(p\). The authors prove the following condition for the vanishing of the \(\lambda\)-invariant of \(L\): we have \(\lambda_p(L)=0\) if and only if the prime ideals \({\mathfrak p}_i\) in \(L_\infty\) that are prime to \(p\) and ramified in \(L_\infty/K_\infty\) generate ideal classes in \(L_\infty\) whose orders are not divisible by \(p\). Applications are given to the special case of such cyclic cubic extensions of \(K=\mathbb{Q}\) for which the splitting fields of the \({\mathfrak p}_i\) are small.