Iwasawa's \(\lambda\)-invariants of certain real quadratic fields (Q915776)

The author continues to study Greenberg's conjecture [\textit{R. Greenberg}, Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)] in the case of a real quadratic field k. Let p be an odd prime number which splits in k. Two natural \(n_ 1\) and \(n_ 2\) are defined from a prime ideal of k lying over p and the fundamental unit of k [see the author and \textit{K. Komatsu}, J. Math. Soc. Japan 38, 95-102 (1986; Zbl 0588.12004) and the author, \textit{K. Komatsu} and \textit{H. Wada}, Proc. Japan Acad., Ser. A 62, 318-319 (1986; Zbl 0612.12004)]. In this note connected closely with the cited papers the case \(n_ 1=n_ 2=2\) is treated further. In particular it is shown that Iwasawa's invariants \(\mu_ p(k)\) and \(\lambda_ p(k)\) both vanish under certain conditions concerning \(n_ 1\), \(n_ 2\) and the p-primary part of the ideal class group of k(exp(2\(\pi\) i/p)).