Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 57: Issue 1
Integral bases of pure fields with square-free parameter
Author: László Remete 1
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  • 1 University of Debrecen, H-4002 Debrecen Pf.400, Hungary
DOI:
https://doi.org/10.1556/012.2020.57.1.1450
Page Count:
91–115
Publication Date:
01 Mar 2020
Online Publication Date:
Mar 2020
Article Category:
Research Article
Full access

Abstract

Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type ℚ(nm) is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases.

In this paper we explicitly give an integral basis of the field ℚ(nm), where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of ℚ(nm) is repeating periodically in m with period length depending on n.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896
Keywords:
Primary 11R04; Integral basis; pure fields; Newton polygons

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