Mathematics of Computation

Author(s): István Gaál; Michael Pohst.
Journal: Math. Comp. 66 (1997), 1689-1696.
MSC (1991): Primary 11Y50; Secondary 11Y40, 11D57
Retrieve article in: PDF
This article is available free of charge

Abstract: We consider the totally real cyclic quintic fields $K_{n}=\mathbb {Q}(\vartheta _{n})$ , generated by a root $\vartheta _{n}$ of the polynomial

$\begin{multline*}f_{n}(x)=x^{5}+n^{2}x^{4}-(2n^{3}+6n^{2}+10n+10)x^{3} +(n^{4}+5n^{3}+11n^{2}+15n+5)x^{2}+(n^{3}+4n^{2}+10n+10)x+1. \end{multline*}$

Assuming that $m=n^{4}+5n^{3}+15n^{2}+25n+25$ is square free, we compute explicitly an integral basis and a set of fundamental units of $K_{n}$ and prove that $K_{n}$ has a power integral basis only for $n=-1,-2$

. For

(both values presenting the same field) all generators of power integral bases are computed.

1.: M.Daberkow, C.Fieker, J.Klüners, M.Pohst, K.Roegner, M.Schörnig & K.Wildanger, Kant V4, J. Symbolic Comp., to appear..
2.: H.Darmon, Note on a polynomial of Emma Lehmer, Math. Comp. 56 (1991), 795-800. MR 91i:11149
3.: I.Gaál, Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comp. 65 (1996), 801-822. MR 96g:11155
4.: I.Gaál, A.Peth\H{o} & M.Pohst, 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. MR 96m:11026
5.: I.Gaál & M.Pohst, On the resolution of index form equations in sextic fields with an imaginary subfield, J.Symbolic Comp., to appear.
6.: I.Gaál and N.Schulte, Computing all power integral bases of cubic number fields, Math. Comp. 53 (1989), 689-696. MR 90b:11108
7.: M.N.Gras, Non monogénéité de l'anneau des entiers des extensions cycliques de ${\mathbb Q}$ de degré premier $l\geq 5$ , J.Number Theory 23 (1986), 347-353. MR 87g:11116
8.: E.Lehmer, Connection between Gaussian periods and cyclic units, Math. Comp. 50 (1988), 535-541. MR 89h:11067a
9.: M.Pohst & H.Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, Cambridge, 1989. MR 92b:11074
10.: R.Schoof & L.Washington, Quintic polynomials and real cyclotomic fields with large class numbers, Math. Comp. 50 (1988), 543-556. MR 89h:11067b
11.: B.W.Char, K.O.Geddes, G.H.Gonnet, M.B.Monagan, S.M.Watt (eds.), MAPLE, Reference Manual, Watcom Publications, Waterloo, Canada, 1988.

Retrieve articles in Mathematics of Computation with MSC (1991): 11Y50, 11Y40, 11D57

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

Michael Pohst
Affiliation: Technische Universität Berlin, Fachbereich 3 Mathematik, Straße des 17. Juni 136, 10623 Germany
Email: pohst@math.tu-berlin.de

DOI: 10.1090/S0025-5718-97-00868-5
PII: S 0025-5718(97)00868-5
Keywords: Power integral basis, family of quintic fields
Received by editor(s): August 13, 1996
Additional Notes: Research supported in part by Grants 16791 and 16975 from the Hungarian National Foundation for Scientific Research and by the Deutsche Forschungsgemeinschaft
Copyright of article: Copyright 1997, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS