Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Power integral bases
in a parametric family
of totally real cyclic quintics

Authors: István Gaál and Michael Pohst
Journal: Math. Comp. 66 (1997), 1689-1696
MSC (1991): Primary 11Y50; Secondary 11Y40, 11D57
Erratum: Math. Comp. 50 (1988), 653.
Erratum: Math. Comp. 41 (1983), 778-779.
MathSciNet review: 1423074
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $n=-1,-2$ (both values presenting the same field) all generators of power integral bases are computed.

References [Enhancements On Off] (What's this?)

  • 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.

Similar Articles

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

Retrieve articles in all journals with MSC (1991): 11Y50, 11Y40, 11D57

Additional Information

István Gaál
Affiliation: Kossuth Lajos University, Mathematical Institute, H–4010 Debrecen Pf.12., Hungary

Michael Pohst
Affiliation: Technische Universität Berlin, Fachbereich 3 Mathematik, Straße des 17. Juni 136, 10623 Germany

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
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society