Open problems on the relation between additive and multiplicative structure



Abstract

This paper presents open problems on the relation between additive and multiplicative structures of cyclotomic fields.


Keywords

unit; normal basis; circulant matrix; cyclotomic field

1. Z. Divišová, J. Kostra, M. Pomp, On transformation matrix connected to normal bases in cubic field, Acta Acad. Paed. Agrriensis, Sec. Mathematicae 27 (2002), to appear.
2. Z. Divišová, J. Kostra, M. Pomp, On transformation matrix connected to normal bases in orders, Journ. of Algebra, Number Theory and Applications, to appear.
3. A. Dvořák, D. Jedelský, J. Kostra, The fields of degree seven over rationals with a normal basis generated by a unit, Math. Slovaca, 49, 2 (1999), 143-153.
4. S. Jakubec, J. Kostra, K. Nemoga, On the existence of an integral normal basis generated by a unit in prime extensions of rational numbers, Math. of Comput., 56, 194 (1991), 809-815.
5. J. Kostra, Orders with a normal basis, Czech. Math. Journ., 35(110) (1985), 391-404.
6. J. Kostra, On sums of two units, Abh. Math. Sem. Univ. Hamburg, 64 (1994), 11-14.
7. M. Newman, Units in arithmetic progression in an algebraic number field, Proc. of the Amer. Math. Soc., 43, 2 (1974), 266-268.
8. M. Newman, Diophantine equations in cyclotomic fields, J. reine angew. Math., 265 (1974), 84-89.
9. M. Newman, Consecutive units, Proc. of the Amer. Math. Soc., 108, 2 (1990), 303-306.
10. M. Newman, Units differing by rationals in a cyclotomic field, Linear and Multilinear Algebra, 34, 1 (1993), 55-57.
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Published : 2003-01-30


KostraJ. (2003). Open problems on the relation between additive and multiplicative structure. Annales Mathematicae Silesianae, 16, 21-25. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14105

Juraj Kostra  kostra@fpv.utc.cz
Department of Applied Mathematics, Faculty of Science, University of Zilina  Slovakia



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