<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>R</italic> be a principal ideal domain with quotient field <italic>K</italic>, and <italic>L</italic> = <italic>K</italic>(<italic>α</italic>), where <italic>α</italic> is a root of a monic irreducible polynomial <italic>F</italic> (<italic>x</italic>) ∈ <italic>R</italic>[<italic>x</italic>]. Let ℤ<italic>L</italic> be the integral closure of <italic>R</italic> in <italic>L</italic>. In this paper, for every prime <italic>p</italic> of <italic>R</italic>, we give a new efficient version of Dedekind’s criterion in <italic>R</italic>, i.e., necessary and sufficient conditions on <italic>F</italic> (<italic>x</italic>) to have <italic>p</italic> not dividing the index [ℤ<italic>L</italic>: <italic>R</italic>[<italic>α</italic>]], for every prime <italic>p</italic> of <italic>R</italic>. Some computational examples are given for <italic>R</italic> = ℤ.</abstract>

Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL be the integral closure of R in L. In this paper, for every prime p of R, we give a new efficient version of Dedekind’s criterion in R, i.e., necessary and sufficient conditions on F (x) to have p not dividing the index [ℤL: R[α]], for every prime p of R. Some computational examples are given for R = ℤ.

Dedekind’s Criterion and Integral Bases

Dept. of Math., Fac. of Sci. Dhar-El Mahraz

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Dedekind\u2019s Criterion and Integral Bases"}

Let R be a principal ideal domain with quotient field K, and L = K(α), where α is a root of a monic irreducible polynomial F (x) ∈ R[x]. Let ℤL be the integral...

Dedekind’s Criterion and Integral Bases

Published Online: Aug 15, 2019

Page range: 1 - 8

Received: Sep 08, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0001

Keywords
Dedekind’s criterion, integral bases, Power integral bases

© 2019 Lhoussain El Fadil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Dedekind’s Criterion and Integral Bases

Published Online: Aug 15, 2019

Page range: 1 - 8

Received: Sep 08, 2018

DOI: https://doi.org/10.2478/tmmp-2019-0001

KeywordsDedekind’s criterion, integral bases, Power integral bases

© 2019 Lhoussain El Fadil, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Dedekind’s criterion, integral bases, Power integral bases