Fukuda, Takashi; Komatsu, Keiichi; Wada, Hideo A remark on the \(\lambda\)-invariant of real quadratic fields. (English) Zbl 0612.12004 Proc. Japan Acad., Ser. A 62, 318-319 (1986). 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 \(n_ 1=n_ 2=2\). Reviewer: Ch.Parry Cited in 2 ReviewsCited in 3 Documents MSC: 11R18 Cyclotomic extensions 11R11 Quadratic extensions 11R23 Iwasawa theory Keywords:\(\lambda \)-invariant; Iwasawa invariants; real quadratic field; class number Citations:Zbl 0588.12004 PDFBibTeX XMLCite \textit{T. Fukuda} et al., Proc. Japan Acad., Ser. A 62, 318--319 (1986; Zbl 0612.12004) Full Text: DOI References: [1] T. Fukuda and K. Komatsu: On the X invariants of Zp-extensions of real quadratic field. J. Number Theory, 23, 238-242 (1986). · Zbl 0593.12003 · doi:10.1016/0022-314X(86)90093-4 [2] T. Fukuda and K. Komatsu: On Zp-extensions of real quadratic fields. J. Math. Soc. Japan, 3£, 95-102 (1986). · Zbl 0588.12004 · doi:10.2969/jmsj/03810095 [3] R. Greenberg: On the Iwasawa invariants of totally real number fields. Amer. J. Math., 98, 263-284 (1976). JSTOR: · Zbl 0334.12013 · doi:10.2307/2373625 [4] S. Maki: The determination of units in real sextic fields. Springer Lecture Notes in Math., vol. 797 (1980). · Zbl 0423.12006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.