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A remark on the \(\lambda\)-invariant of real quadratic fields. (English) Zbl 0612.12004

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

MSC:

11R18 Cyclotomic extensions
11R11 Quadratic extensions
11R23 Iwasawa theory

Citations:

Zbl 0588.12004
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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
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