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Power Integral Bases in Composits of Number Fields

Published online by Cambridge University Press:  20 November 2018

István Gaál
István Gaál*
Affiliation:
Kossuth Lajos University Mathematical Institute H-4010 Debrecen Pf.12. Hungary, e-mail: igaal@math.klte.hu
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Abstract

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In the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

Keywords

11D57:11R21
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 41 , Issue 2 , 01 June 1998 , pp. 158 - 165
Copyright
Copyright © Canadian Mathematical Society 1998

References

1. Daberkow, M., Fieker, C., Klüners, J., Pohst, M., Roegner, K., Schörnig, M. and Wildanger, K., Kant V4. J. Symb. Comput. 24 (1997), 267–283.Google Scholar
2. Darmon, H., Note on a polynomial of Emma Lehmer. Math. Comp. 56 (1991), 795–800.Google Scholar
3. Gaál, I., Computing all power integral bases in orders of totally real cyclic sextic number fields. Math. Comp. 65 (1996), 801–822.Google Scholar
4. Gaál, I., Computing elements of given index in totally complex cyclic sextic fields. J. Symb. Comput., 20 (1995), 61–69.Google Scholar
5. Gaál, I., Pethőo, A. and Pohst, M., On the resolution of index form equations in quartic number fields. J. Symb. Comput., 16 (1993), 563–584.Google Scholar
6. Gaál, I., Simultaneous representation of integers by a pair of ternary quadratic forms—with an application to index form equations in quartic number fields. J. Number Theory 57 (1996), 90–104.Google Scholar
7. Gaál, I. and Pohst, M., On the resolution of index form equations in sextic fields with an imaginary quadratic subfield. J. Symb. Comput. 22 (1996), 425–434.Google Scholar
8. Gaál, I., Power integral bases in a parametric family of totally real cyclic quintics. Math. Comp. 66 (1997), 1689–1696.Google Scholar
9. Gaál, I. and Schulte, N., Computing all power integral bases of cubic number fields. Math. Comp. 53 (1989), 689–696.Google Scholar
10. Gras, M. N., Non monogénéité de l’anneau des entiers des extensions cycliques de Q de degré premier l½ 5. J. Number Theory, 23 (1986), 347–353.Google Scholar
11. Lehmer, E., Connection between Gaussian periods and cyclic units. Math. Comp. 50 (1988), 535–541.Google Scholar
12. Narkiewicz, W., Elementary and Analytic Theory of Algerbaic Numbers. Second Edition, SpringerVerlag, 1990.Google Scholar
13. Schoof, R. and Washington, L., Quintic polynomials and real cyclotomic fields with large class numbers. Math. Comp. 50 (1988), 543–556.Google Scholar