Jhorar, Bablesh ; Khanduja, Sudesh K.
(2016)
*On power basis of a class of algebraic number fields*
International Journal of Number Theory, 12
(08).
Article ID 2317.
ISSN 1793-0421

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Official URL: http://www.worldscientific.com/doi/abs/10.1142/S17...

Related URL: http://dx.doi.org/10.1142/S1793042116501384

## Abstract

Let K=Q(θ) be an algebraic number field with θ in the ring A_{K} of algebraic integers of K and F(x) be the minimal polynomial of θ over the field Q of rational numbers. In 1977, Uchida proved that A_{K}=Z[θ] if and only if F(x) does not belong to M^{2} for any maximal ideal M of the polynomial ring Z[x] (see [Osaka J. Math. 14 (1977) 155–157]). In this paper, we apply the above result to obtain some necessary and sufficient conditions involving the coefficients of F(x) for AK to equal Z[θ] when F(x) is a trinomial of the type x^{n}+ax+b. In the particular case when a=−1, it is deduced that {1,θ,…,θ^{n−1}} is an integral basis of K if and only if either (i) p∤b and p^{2}∤(b^{n−1}n^{n}−(n−1)^{n−1}) or (ii) p divides b and p^{2}∤b.

Item Type: | Article |
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Source: | Copyright of this article belongs to World Scientific Publishing Company. |

Keywords: | Rings of Algebraic Integers; Integral Basis and Discriminant |

ID Code: | 112143 |

Deposited On: | 23 Jan 2018 12:19 |

Last Modified: | 23 Jan 2018 12:19 |

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