×

Normal integral basis of an unramified quadratic extension over a cyclotomic \(\mathbb{Z}_2\)-extension. (English. French summary) Zbl 1358.11120

Let \(L/K\) be a finite Galois extension of number fields with Galois group \(G\). Noether showed that the ring of integers \(\mathcal{O}_{L}\) is projective as a module over the group ring \(\mathcal{O}_{K}[G]\) if and only if \(L/K\) is at most tamely ramified. The extension \(L/K\) is said to have a normal integral basis (NIB) if \(\mathcal{O}_{L}\) is in fact free (necessarily of rank \(1\)) over \(\mathcal{O}_{K}[G]\).
J. Brinkhuis [Bull. Lond. Math. Soc. 24, 236–242 (1992; Zbl 0768.11045)] showed that if \(K\) is a totally real number field then a finite abelian extension \(L/K\) that is unramified at all finite primes never has a NIB, with the possible exception of a multi-quadratic extension of \(K\). The article under review considers this possible exception in a special case.
Let \(K/\mathbb{Q}\) be a cyclic extension of odd prime degree \(\ell\). Let \(K_{n}\) denote the \(n\)th layer of the cyclotomic \(\mathbb{Z}_{2}\)-extension \(K_{\infty}/K\) with \(K_{0}=K\). Note that \(K_{n}\) is totally real for every \(n \geq 0\). Let \(A_{n}\) be the Sylow \(2\)-subgroup of the ideal class group of \(K_{n}\) and let \(H/K\) be the class field corresponding to the quotient \(A_{0}/A_{0}^{2}\). The article under review considers the following two questions:
(Q1) Does \(H/K\) have a NIB?
(Q2) If not, does the pushed-up extension \(HK_{n}/K_{n}\) have a NIB for some \(n \geq 1\)?
The authors work under several assumptions, including:
(A1) The prime number \(2\) is a primitive root modulo \(\ell\).
(A2) The prime number \(2\) remains prime in \(K\).
Let \(\zeta_{\ell}\) denote a primitive \(\ell\)-root of unity and let \(\mathbb{Q}_{2}\) denote the field of \(2\)-adic rationals. We note that (A1) is equivalent to \([\mathbb{Q}_{2}(\zeta_{\ell}):\mathbb{Q}_{2}] = \ell -1\). Let \(\Delta=\mathrm{Gal}(K/\mathbb{Q})\). Fix a non-trivial \(\overline{\mathbb{Q}}_{2}\)-valued character \(\chi\) of \(\Delta\) and let \(\chi_{0}\) denote the trivial character of \(\Delta\). Assumption (A1) means that the \(2\)-adic group ring \(\mathbb{Z}_{2}[\Delta]\) decomposes into components corresponding to \(\chi_{0}\) and \(\chi\) and thus any \(\mathbb{Z}_{2}[\Delta]\)-module \(M\) decomposes as \(M=M(\chi_{0}) \oplus M(\chi)\). It is well known that \(A_{n}(\chi_{0})\) is trivial for all \(n \geq 0\) and hence one can deduce that \(|A_{0}|=|A_{0}(\chi)|=2^{\kappa(\ell-1)}\) for some \(\kappa \geq 0\). Let \(\mathcal{O}_{\chi}=\mathbb{Z}_{2}[\zeta_{\ell}]\) be the \(\mathbb{Z}_{2}\)-algebra generated by values of \(\chi\). Let \(P_{\chi}(t) \in \mathcal{O}_{\chi}[t]\) be the distinguished polynomial associated to the power series arising from the \(2\)-adic \(L\)-function \(L_{2}(s,\chi)\) and let \(\lambda_{\chi}=\deg P_{\chi}\). The authors make the following further assumption:
(A3) \(A_{0} \cong \mathcal{O}_{\chi}/2^{\kappa}\mathcal{O}_{\chi}\) with some \(\kappa \geq 1\).
Under assumptions (A1)–(A3), the authors give the following answer to (Q1): \(H/K\) has a NIB if and only if \(P_{\chi}(0) \equiv 0 \bmod 2^{\kappa+1}\).
To deal with (Q2), the authors make two further assumptions:
(A4) \(2^{\kappa} \parallel P_{\chi}(0)\).
(A5) \(|A_{0}|=|A_{1}|\).
Under assumptions (A1)–(A5), the authors consider (Q2) in four cases, depending on whether \(\lambda_{\chi}=1\) or \(\lambda_{\chi} \geq 2\) and whether \(\kappa=1\) or \(\kappa \geq 2\).
Finally, the authors give numerical examples in the case \(\ell=3\); more precisely, they consider cyclic cubic fields \(K\) of prime conductor \(p\) with \(p \equiv 1 \bmod 3\) and \(p< 10^{4}\).

MSC:

11R20 Other abelian and metabelian extensions
11R23 Iwasawa theory

Citations:

Zbl 0768.11045
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] KANT/Kash3, .
[2] UBASIC, (in Japanese). · Zbl 0807.11060
[3] J. Brinkhuis, « Unramified abelian extensions of CM-fields and their Galois module structure », Bull. London Math. Soc.24 (1992), no. 3, p. 236-242. · Zbl 0768.11045
[4] A. Brumer, « On the units of algebraic number fields », Mathematika14 (1967), p. 121-124. · Zbl 0171.01105
[5] L. N. Childs, « The group of unramified Kummer extensions of prime degree », Proc. London Math. Soc. (3)35 (1977), no. 3, p. 407-422. · Zbl 0374.13002
[6] A. Fröhlich & M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27, Cambridge University Press, Cambridge, 1993, xiv+355 pages. · Zbl 0744.11001
[7] T. Fukuda, « Remarks on \(\mathbf{Z}_p\)-extensions of number fields », Proc. Japan Acad. Ser. A Math. Sci.70 (1994), no. 8, p. 264-266. · Zbl 0823.11064
[8] R. Gillard, « Unités cyclotomiques, unités semi-locales et \({\bf Z}_l\)-extensions. II », Ann. Inst. Fourier (Grenoble)29 (1979), no. 4, p. viii, 1-15. · Zbl 0403.12006
[9] M.-N. Gras, « Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de \({\bf Q}\) », J. Reine Angew. Math.277 (1975), p. 89-116. · Zbl 0315.12007
[10] H. Ichimura, « On \(p\)-adic \(L\)-functions and normal bases of rings of integers », J. Reine Angew. Math.462 (1995), p. 169-184. · Zbl 0815.11055
[11] —, « On a normal integral bases problem over cyclotomic \({\bf Z}_p\)-extensions », J. Math. Soc. Japan48 (1996), no. 4, p. 689-703. · Zbl 0892.11036
[12] —, « Class number parity of a quadratic twist of a cyclotomic field of prime power conductor », Osaka J. Math.50 (2013), no. 2, p. 563-572.
[13] —, « Semi-local units at \(p\) of a cyclotomic \(\mathbb{Z}_p\)-extension congruent to \(1\) modulo \(\zeta_p-1\) », Hokkaido Math. J.44 (2015), p. 397-407.
[14] —, « On a duality of Gras between totally positive and primary cyclotomic units », Math. J. Okayama Univ.58 (2016), p. 125-132. · Zbl 1408.11109
[15] H. Ichimura, S. Nakajima & H. Sumida-Takahashi, « On the Iwasawa lambda invariant of an imaginary abelian field of conductor \(3p^{n+1}\) », J. Number Theory133 (2013), no. 2, p. 787-801. · Zbl 1286.11175
[16] H. Ichimura & H. Sumida, « A note on integral bases of unramified cyclic extensions of prime degree. II », Manuscripta Math.104 (2001), no. 2, p. 201-210. · Zbl 0991.11058
[17] F. Kawamoto & Y. Odai, « Normal integral bases of \(\infty \)-ramified abelian extensions of totally real number fields », Abh. Math. Sem. Univ. Hamburg72 (2002), p. 217-233. · Zbl 1036.11058
[18] J. S. Kraft & R. Schoof, « Computing Iwasawa modules of real quadratic number fields », Compositio Math.97 (1995), no. 1-2, p. 135-155, Special issue in honour of Frans Oort. · Zbl 0840.11043
[19] W. Narkiewicz, Elementary and analytic theory of algebraic numbers, third ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004, xii+708 pages. · Zbl 1159.11039
[20] B. Oriat, « Relation entre les \(2\)-groupes des classes d’idéaux au sens ordinaire et restreint de certains corps de nombres », Bull. Soc. Math. France104 (1976), no. 3, p. 301-307. · Zbl 0352.12007
[21] W. Sinnott, « On the Stickelberger ideal and the circular units of an abelian field », Invent. Math.62 (1980/81), no. 2, p. 181-234. · Zbl 0465.12001
[22] A. Srivastav & S. Venkataraman, « Relative Galois module structure of quadratic extensions », Indian J. Pure Appl. Math.25 (1994), no. 5, p. 473-488. · Zbl 0804.11063
[23] —, « Unramified quadratic extensions of real quadratic fields, normal integral bases, and \(2\)-adic \(L\)-functions », J. Number Theory67 (1997), no. 2, p. 139-145. · Zbl 0891.11052
[24] M. Taylor, « Galois module structure of classgroups and units », Mathematika22 (1975), no. 2, p. 156-160. · Zbl 0322.12010
[25] —, « The Galois module structure of certain arithmetic principal homogeneous spaces », J. Algebra153 (1992), no. 1, p. 203-214. · Zbl 0776.11065
[26] L. C. Washington, Introduction to cyclotomic fields, second ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997, xiv+487 pages. · Zbl 0966.11047
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.