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The Iwasawa invariant \(\mu\) vanishes for \(\mathbb{Z}_{2}\)-extensions of certain real biquadratic fields

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Abstract

For a real biquadratic field, we denote by \(\lambda\), \(\mu\) and \(\nu\) the Iwasawa invariants of cyclotomic \(\mathbb{Z}_{2}\)-extension of \(k\). We give certain families of real biquadratic fields \(k\) such that \(\mu=0\).

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References

  1. A. Azizi and A. Mouhib, Capitulation des 2-classes d’idéaux de certains corps biquadratiques dont le corps de genres diffère du 2-corps de classes de Hilbert, Pacific J. Math., 218 (2005), 17–36.

  2. A. Azizi and M. Talbi, Capitulation des 2-classes d’idéaux de certains corps biquadratiques cycliques, Acta Arith., 127 (2007), 231–248.

  3. T. Fukuda, Remarks on \(\mathbb{Z}_{p}\)-extensions of number fields, Proc. Japan Acad. Ser. A, 70 (1994), 264–266.

  4. G. Gras, Sur les l-classes d’idéaux dans les extensions cycliques relatives de degr´e premier l. 1, Ann. Inst. Fourier (Grenoble), 23 (1973), 1–48.

  5. R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math., 98 (1976), 263–284.

  6. H. Hasse, Neue Begründung und Verallgemeinerung der Theorie des Normenrestsymbols, J. Reine Angew. Math., 162 (1930), 134–44.

  7. P. Kaplan, Sur le 2-groupe des classes d’idéaux des corps quadratiques, J. Reine Angew. Math., 283/284 (1976), 313–363.

  8. G. Karpilovsky, Unit Groups of Classical Rings, Oxford University Press (1988).

  9. T. Kubota, Über den bizyklischen biquadratischen Zahlkörper, Nagoya Math. J., 10 (1956), 65–85.

  10. A. Mouhib, On the parity of the class number of multiquadratic number fields, J. Number Theory, 129 (2009), 1205–1211.

  11. M. Saito and H. Wada, Tables of ideal class group of real quadratic fields, Proc. Japan Acad., 64 (1988), 347–349.

  12. L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Graduate Texts in Math., vol. 83, Springer (1997).

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Acknowledgement

The authors thank the managing editor and anonymous referee for valuable comments which helped them significantly to consolidate the paper.

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  1. ACSA Laboratory, Department of Mathematics, Faculty of Sciences, Mohammed First University, Oujda, Morocco

    A. EL Mahi & M. Ziane

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  1. A. EL Mahi
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  2. M. Ziane
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Correspondence to A. EL Mahi.

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EL Mahi, A., Ziane, M. The Iwasawa invariant \(\mu\) vanishes for \(\mathbb{Z}_{2}\)-extensions of certain real biquadratic fields. Acta Math. Hungar. 165, 146–155 (2021). https://doi.org/10.1007/s10474-021-01174-2

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  • DOI: https://doi.org/10.1007/s10474-021-01174-2

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