Classes de steinitz et extensions quaternioniennes
DOI:
https://doi.org/10.22199/S07160917.1997.0001.00001Keywords:
Stetnitz clases, Explicit embedding problema, Tamely or wildly ramified quaternionic extensionsAbstract
Let k be a number field. We prove that the set of Steinitz clases of quadratic extensions K/ k which can be embedded in quaternionic extensions of degree 8, tamely or wildly ramified over k, is the full class group Cl(k) of k. Moreover, we provide explicit formulas to construct these quaternionic extensions.
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References
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[13] E. ROSENBLÜTH. Die arithmetische Theorie und Konstruktion der Quaternionenkörper auf Klassenkörpertheoretischer Grundlage, Monatsh. Math. Phys. 41, pp. 85-125, (1934).
[14] .J.-P. SEHHE. Corps Locaux , 3è éd., Hermann, Paris, (1980).
[15] B. SOGAÏGUI, Structure galoisienne relative des anneaux d'entiers , .J. Number Theory 28, pp. 189-204, (1988).
[16] E. WITT, Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf, .J. Crelle 174, pp. 237-245, (1936).
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[2] P. DAMEY et J. MARTINET, Plongement d'une extension quadratique dans ane extension quaternionienne, J. reine angew. Math. 262-263. pp. 323-338. (1973).
[3] A. FRÖHLICH. The discriminants of relative extensions and the existence of inteqral bases, Mathematika 7, pp. 15-22, (1960).
[4] A. FRÖHLICH, Artin-rool numbers and normal integral bases for quaternion fields, Invent. Math. 17, pp. 143-166, (1972).
[5] C. FU.JISAKT. An elementary construction of Galois quaternion extension. Pruc . .Japan Acad. 66A, pp. 80-83, (1990).
[6] E. HECKE, Lectures on the Theory of Algebraic Numbers, Graduate Texts Math. 77. Springer-Verlag, New York, (1981).
[7] S. LANG. Algebraic Numbcr Theory. Graduate Texts Math. 110, Springer-Verlag. New York, 1986.
[8] R. LONG. Steiwitz classes of eyelie extensions of degree 1r, Proc. Amer. Mlath. Suc. 49. pp. 297-304, (1975).
[9] J. MARTINET. Les discriminants quadratiques et la congruence de Stickelberger. Sém. Théor. Nombres Bordeaux 1, pp. 197-204, (1989).
[10] R. MASSY. Construction de p-extensions galoisiennes d'un corps de caracteristique différente de p. J. Algebra 109. pp. 508-535, (1987).
[11] R. MASSY. Bases normales d 'entiers relatives quadratiques, .J. Number Theory 38, pp. 216-239, (1991).
[12] J. NEUKIRCH. Über das Einbettunqspmblem der algebraischen Zahlentheorie. Invent. Math. 21, pp. 59-116, (1973).
[13] E. ROSENBLÜTH. Die arithmetische Theorie und Konstruktion der Quaternionenkörper auf Klassenkörpertheoretischer Grundlage, Monatsh. Math. Phys. 41, pp. 85-125, (1934).
[14] .J.-P. SEHHE. Corps Locaux , 3è éd., Hermann, Paris, (1980).
[15] B. SOGAÏGUI, Structure galoisienne relative des anneaux d'entiers , .J. Number Theory 28, pp. 189-204, (1988).
[16] E. WITT, Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf, .J. Crelle 174, pp. 237-245, (1936).
[17] K.S. WILLIAMS, Integers of biquadratic fields, Canad. Math. Bull. 13(4). pp. 519-526. (1970).
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2018-04-04
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How to Cite
[1]
“Classes de steinitz et extensions quaternioniennes”, Proyecciones (Antofagasta, On line), vol. 16, no. 1, pp. 1–13, Apr. 2018, doi: 10.22199/S07160917.1997.0001.00001.
