A remark on the \(\lambda\)-invariant of real quadratic fields
Sufficient conditions are given for the Iwasawa invariants of a real quadratic field, k, to satisfy \(\lambda_ p(k)=\mu_ p(k)=0\) when \(n_ 1=n_ 2=2\). In previous work of the first two of the authors [J. Math. Soc. Japan 38, 95-102 (1986; Zbl 0588.12004)], it was required that \(n_ 1<n_ 2\). Let h and \(\epsilon\) denote the class number and fundamental unit of k, respectively. If p is an odd prime which splits completely in k and \({\mathfrak p}\) is a prime divisor of p in k then \({\mathfrak p}^ h=(\alpha)\) for some \(\alpha\in k\). Then \(n_ 1\) (resp. \(n_ 2)\) is defined to be the maximal integer such that \(\alpha^{p-1}\equiv 1\) (mod \(p^{n_ 1} {\mathbb{Z}}_ p)\), (resp. \(\epsilon^{p-1}\equiv 1\) (mod \(p^{n_ 2} {\mathbb{Z}}_ p))\). Using their criteria, the authors give eleven fields with \(\lambda_ 3(k)=\mu_ 3(k)=0\) and \(n_ 1=n_ 2=2\).
scientific article; zbMATH DE number 3989479