Cougnard, Jean Stably free rings of integers over \(\mathbb{Z}[H_8\times C_2]\). (Anneaux d’entiers stablement libres sur \(\mathbb{Z}[H_8\times C_2]\).) (French) Zbl 0924.11093 J. Théor. Nombres Bordx. 10, No. 1, 163-201 (1998). Let \(G= H_8\times C_2\), the direct product of the quaternion group of order 8 with a cyclic group of order 2. It is known that there are four isomorphism classes of non-free stably free \(\mathbb{Z} G\)-modules. The aim of this paper is to show that each of these four classes is represented infinitely often by rings of integers \({\mathcal O}_N\), where \(N\) is a tamely ramified Galois extension of \(\mathbb{Q}\) with Galois group \(G\). This bald statement of the result hides the immense amount of calculation that is needed; these calculations will be of independent interest to anyone who whishes to compute explicitly with quaternionic and related extensions of the rational numbers. Sections I to V detail the method of constructing quaternionic and \(G\)-extensions of \(\mathbb{Q}\), together with an analysis of the ambiguous ideals in such extensions and their quadratic and biquadratic subfields. The methods are adaptions of those used by Witt and Martinet. Section VI reviews the computation of the locally free class group of \(\mathbb{Q} G\), and identifies the isomorphism classes of non-free stably free \(\mathbb{Z} G\)-modules in terms of units of the local ring \(\mathbb{F}_2 H_8\), thus recovering results due to Swan. These ingredients are combined in section VII to produce explicit infinite families of extensions, as promised. Finally, the author considers a numerical example that contains the field \(\mathbb{Q} (\sqrt{1001}, \sqrt{2805})\). Reviewer: M.E.Keating (London) Cited in 5 Documents MSC: 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers 20C10 Integral representations of finite groups 11R20 Other abelian and metabelian extensions 19A13 Stability for projective modules Keywords:quaternionic extension; biquadratic bicyclic extension; non-free stably free module; class group; representations by rings of algebraic integers; ambiguous ideals Software:PARI/GP × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML EMIS References: [1] Chevalley, C., Sur certains idéaux dans une algèbre simple, Abh. Math. Sem. Univ. Hamburg10 (1934), 83-105. · JFM 60.0101.02 [2] Cougnard, J., Un anneau d’entiers stablement libre et non libre, Experimental Mathematics3 n°2 (1994), 129-136. · Zbl 0837.11062 [3] Crespo, T., explicit construction of Ãn Type fields, J. of Algebra127 n° 2 (1989), 452-461. · Zbl 0704.11043 [4] Eichler, M., Über die Idealklassenzahl in gewissen normalen einfachenAlgebren, Math. Zeit.43 (1938), 481-494. · JFM 64.0085.01 [5] Fröhlich, A., Artin-Root Numbers and Normal Integral Bases for Quaternion fields, Inventiones Math.17 (1972), 143-166. · Zbl 0261.12008 [6] Fröhlich, A., Locally free modules over arithmetic orders, J. reine angew. Math274/75 (1975), 112-138. · Zbl 0316.12013 [7] Fröhlich, A., Arithmetic and Galois module structure for tame extensions, J. reine angew. Math286/287 (1976), 380-440. · Zbl 0385.12004 [8] Frôhlich, A., Galois module structure of agebraic integers, Springer Verlag (1983). · Zbl 0501.12012 [9] Jacobinski, H., Genera and decomposition of lattices over orders, Acta Math121 (1968), 1-29. · Zbl 0167.04503 [10] Keating, M.E., On the K-theory of the quaternion group, Mathematika20 (1973), 59-62. · Zbl 0267.18016 [11] Lam, T.Y., The algebraic theory of quadratic forms, seconde édition, Benjamin (1980). · Zbl 0437.10006 [12] Martinet, J., Modules sur l’algébre du groupe quaternionien, Annales Sci. de l’Ec. normale sup 4 série fasc. 3 (1971), 399-408. · Zbl 0219.12012 [13] Martinet, J., Sur les extensions à groupe de Galois quaternionien, C.R. Acad. Sc. Paris t. 274 (1972), 933-935. · Zbl 0235.12005 [14] Martinet, J., H8, Algebraic Number Fields, 525-538 édité par A. Fröhlich, Academic Press1977. · Zbl 0359.12014 [15] Milnor, J., Introduction to algebraic K-Theory, Annals of Math. Studies72 (1971). · Zbl 0237.18005 [16] Massy, R., Do, T. Nuyen Quang, Plongement d’une extension de degré p2 dans une surextension non abélienne de degré p3: étude locale-globale, J. für die reine und angew. Math. t. 291 (1977), 149-161. · Zbl 0346.12005 [17] Batut, C., Bernardi, D., Cohen, H., Olivier, M., User’s Guide to Pari-GP, version 1.39-12 (1995). [18] Serre, J-P., Modules projectifs et espaces fibrés à fibres vectorielles, Séminaire Dubreil exposén° 23 (1968), 23.01-23.18. · Zbl 0132.41202 [19] Swan, R.G., Induced representations and projective modules, Ann. of Math.71 (1960), 552-578. · Zbl 0104.25102 [20] Swan, R.G., Projective modules over group rings and maximal orders, Ann. of Math.76 (1962), 55-61. · Zbl 0112.02702 [21] Swan, R.G., Strong approximation theorem and locally free modules, dans Ring Theory and Algebra III ed. (1980). · Zbl 0478.12010 [22] Swan, R.G., Projective modules over binary polyhedral groups, J. reine angew. Math.342 (1983), 66-172. · Zbl 0503.20001 [23] Taylor, M.J., On Fröhlich’s conjecture for rings of integers of tame extensions, Invent. Math.63 (1981), 41-79. · Zbl 0469.12003 [24] Ullom, S., Normal bases in Galois extensions of number fields, Nagoya Math. J.34 (1969), 153-167. · Zbl 0175.04502 [25] Witt, E., Konstruktion von Körpern zu vorgegebener gruppe der ordnung pf, J. reine angew. Math.174 (1936), 237-245. · JFM 62.0110.02 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.