Abstract
Let m ≠ 0 be an integer which is not a perfect square and consider number fields of the form \(\mathbb{Q}\left[ {\sqrt[4]{m}} \right]\). We characterize all orders of the form \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\) which admit a unit power integral basis, i.e., there exists a unit ε such that 1, ε, ε 2 and ε 3 is an integral basis of \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Ashrafi and P. Vámos, On the unit sum number of some rings, Q. J. Math., 56 (2005), 1–12.
P. Belcher, Integers expressible as sums of distinct units, Bull. Lond. Math. Soc., 6 (1974), 66–68.
P. Belcher, A test for integers being sums of distinct units applied to cubic fields, J. Lond. Math. Soc. (2), 12 (1976), 141–148.
M. A. Bennett, Rational approximation to algebraic numbers of small height: the Diophantine equation |ax n − by n| = 1, J. Reine Angew. Math., 535 (2001), 1–49.
M. A. Bennett and B. M. M. Deweger, On the Diophantine equation |ax n−by n| = 1, Math. Comp., 67 (1998), 413–438.
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235–265.
H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer, Berlin, 1993.
A. Filipin, R. Tichy and V. Ziegler, The additive unit structure of purely quartic complex fields, Funct. Approx. Comment. Math., 39 (2008), 113–131.
I. Gaál, Solving index form equations in fields of degree 9 with cubic subfields, J. Symbolic Comput., 30 (2000), 181–193.
I. Gaál, Diophantine equations and power integral bases, Birkhäuser Boston Inc., Boston, MA, 2002.
I. Gaál, P. Olajos and M. Pohst, Power integral bases in orders of composite fields, Experiment. Math., 11 (2002), 87–90.
I. Gaál, A. Pethő and M. Pohst, On the resolution of index form equations in biquadratic number fields. III. The bicyclic biquadratic case, J. Number Theory, 53 (1995), 100–114.
I. Gaál and N. Schulte, Computing all power integral bases of cubic fields, Math. Comp., 53 (1989), 689–696.
K. Győry, Bounds for the solutions of norm form, discirminant form and index form equations in finitely generated integral domains, Acta Math. Hungar., 42 (1983), 45–80.
K. Győry, Sur les polynômes à coefficients entiers et de discriminant donné. III, Publ. Math. Debrecen, 23 (1976), 141–165.
B. Jacobson, Sums of distinct divisors and sums of distinct units, Proc. Am. Math. Soc., 15 (1964), 179–183.
M. Jarden and W. Narkiewicz, On sums of units, Monatsh. Math., 150 (2007), 327–332.
W. Ljunggren, Eigenschaften der Einheiten reeller quadratischer und rein biquadratischer Zahlkörper mit Anwendung auf die Lösung einer Klasse unbestimmter Gleichungen vierten Grades, Skr. Norske Vid.-Akad., 12 (1936), 1–73.
C. L. Siegel, Die Gleichung ax n − by n = c, Math. Ann., 114 (1937), 57–68.
J. ŚLIWA, Sums of distinct units, Bull. Acad. Pol. Sci., 22 (1974), 11–13.
A. Thue, Berechnung aller Lösungen gewisser Gleichungen von der Form ax r−by r = f, Vid. Skrivter I Mat.-Naturv. Klasse, 4 (1918), 1–9.
R. Tichy and V. Ziegler, Units generating the ring of integers of complex cubic fields, Colloq. Math., 109 (2007), 71–83.
A. Togbé, P. M. Voutier and P. G. Walsh, Solving a family of Thue equations with an application to the equation x 2−Dy 4 = 1, Acta Arith., 120 (2005), 39–58.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Attila Pethő
The author was supported by the Austrian Sience Found (FWF) under the project J2886-NT.
Rights and permissions
About this article
Cite this article
Ziegler, V. On unit power integral bases of \(\mathbb{Z}\left[ {\sqrt[4]{m}} \right]\) . Period Math Hung 63, 101–112 (2011). https://doi.org/10.1007/s10998-011-7101-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-011-7101-9