Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-15

On tamely ramified Iwasawa module of real quadratic fields

Karim BOULAJHAF, Ali MOUHIB

Received June 19, 2025. Published online March 11, 2026.

Abstract: Let $q$ be a prime number such that $q\equiv5$ $({\rm mod} 8)$ and $S=\lbrace q\rbrace$. Let $k$ be a real quadratic number field, and $k_\infty$ its cyclotomic $\mathbb Z_2$-extension. We study the cyclicity of the Galois group $X'_{\infty,S}$ over $k_\infty$ of the maximal $S$-ramified 2-split abelian 2-extension. As a consequence, we determine the complete list of real quadratic number fields for which $X'_{\infty,S}$ is cyclic. Furthermore, we give an infinite family of real quadratic number fields for which $X'_{\infty,S}$ is finite.

Keywords: Iwasawa theory; 2-rank; 2-ray class group; real quadratic fields

Classification MSC: 11R23, 11R37

DOI: 10.21136/MB.2026.0092-25

PDF available at: Institute of Mathematics CAS

References:
[1] T. Fukuda: Remarks on $\Bbb{Z}_p$-extensions of number fields. Proc. Japan Acad., Ser. A 70 (1994), 264-266. DOI 10.3792/pjaa.70.264 | MR 1303577 | Zbl 0823.11064
[2] G. Gras: Class Field Theory: From Theory to Practice. Springer Monographs in Mathematics. Springer, Berlin (2003). DOI 10.1007/978-3-662-11323-3 | MR 1941965 | Zbl 1019.11032
[3] K. Iwasawa: On $\Bbb{Z}_{\ell}$-extensions of algebraic number fields. Ann. Math. (2) 98 (1973), 246-326. DOI 10.2307/1970784 | MR 0349627 | Zbl 0285.12008
[4] A. Mouhib, A. Movahhedi: Cyclicity of the unramified Iwasawa module. Manuscr. Math. 135 (2011), 91-106. DOI 10.1007/s00229-010-0407-8 | MR 2783388 | Zbl 1234.11150
[5] M. Ozaki, G. Yamamoto: Iwasawa $\lambda_3$-invariants of certain cubic fields. Acta Arith. 97 (2001), 387-398. DOI 10.4064/aa97-4-7 | MR 1823554 | Zbl 1059.11064
[6] L. Salle: On maximal tamely ramified pro-2-extensions over the cyclotomic $\Bbb{Z}_2$-extension of an imaginary quadratic field. Osaka J. Math. 47 (2010), 921-942. MR 2791570 | Zbl 1263.11097

Affiliations: Karim Boulajhaf (corresponding author), Ali Mouhib, University Mohammed Ben Abdellah Fes, Mathematics and Data Science Laboratory (MDSL), Polydisciplinary Faculty of Taza, B. P. 1223, Taza-Gare, Morocco, e-mail: boulajhafk07@gmail.com, mouhibali@yahoo.fr

[List of online first articles] [Contents of Mathematica Bohemica] [Full text of the older issues of Mathematica Bohemica at DML-CZ]

PDF available at:

Institute of Mathematics CAS
free access