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NORMAL BASES FOR MODULAR FUNCTION FIELDS
Published online by Cambridge University Press: 02 March 2017
Abstract
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We provide a concrete example of a normal basis for a finite Galois extension which is not abelian. More precisely, let $\mathbb{C}(X(N))$ be the field of meromorphic functions on the modular curve $X(N)$ of level $N$. We construct a completely free element in the extension $\mathbb{C}(X(N))/\mathbb{C}(X(1))$ by means of Siegel functions.
MSC classification
Secondary:
11G16: Elliptic and modular units
- Type
- Research Article
- Information
- Copyright
- © 2017 Australian Mathematical Publishing Association Inc.
Footnotes
The second author was supported by Hankuk University of Foreign Studies Research Fund of 2016.
References
Blessenohl, D. and Johnsen, K., ‘Eine Verschärfung des Satzes von der Normalbasis’, J. Algebra
103(1) (1986), 141–159.CrossRefGoogle Scholar
Hachenberger, D., ‘Universal normal bases for the abelian closure of the field of rational numbers’, Acta Arith.
93(4) (2000), 329–341.CrossRefGoogle Scholar
Jung, H. Y., Koo, J. K. and Shin, D. H., ‘Normal bases of ray class fields over imaginary quadratic fields’, Math. Z.
271(1–2) (2012), 109–116.Google Scholar
Koo, J. K. and Shin, D. H., ‘Completely normal elements in some finite abelian extensions’, Cent. Eur. J. Math.
11(10) (2013), 1725–1731.Google Scholar
Kubert, D. and Lang, S., Modular Units, Grundlehren der Mathematischen Wissenschaften, 244 (Springer, New York, 1981).Google Scholar
Lang, S., Elliptic Functions, 2nd edn, Graduate Texts in Mathematics, 112 (Springer, New York, 1987).CrossRefGoogle Scholar
Leopoldt, H.-W., ‘Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers’, J. reine angew. Math.
201 (1959), 119–149.CrossRefGoogle Scholar
Okada, T., ‘On an extension of a theorem of S. Chowla’, Acta Arith.
38(4) (1980/81), 341–345.Google Scholar
Schertz, R., ‘Galoismodulstruktur und elliptische Funktionen’, J. Number Theory
39(3) (1991), 285–326.CrossRefGoogle Scholar
Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions (Iwanami Shoten and Princeton University Press, Princeton, NJ, 1971).Google Scholar
Taylor, M. J., ‘Relative Galois module structure of rings of integers and elliptic functions II’, Ann. of Math. (2)
121(3) (1985), 519–535.CrossRefGoogle Scholar
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