<abstract xmlns="http://www.w3.org/1999/xhtml">

Let <italic>K </italic>be a number field, and let <italic>ℓ</italic> be a prime number. Fix some elements <italic>α</italic>1,...<italic>,αr</italic> of <italic>K× </italic>which generate a subgroup of <italic>K× </italic>of rank <italic>r</italic>. Let <italic>n</italic>1,...<italic>,nr</italic>, <italic>m</italic> be positive integers with <italic>m</italic> ⩾ <italic>ni</italic> for every <italic>i</italic>. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension <italic>K</italic>(<italic>ζ</italic>ℓm, 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2021-0008_ineq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mroot><mrow><msub><mi>α</mi><mn>1</mn></msub></mrow><mrow><msup><mi>ℓ</mi><mrow><msub><mi>n</mi><mn>1</mn></msub></mrow></msup></mrow></mroot><mo>,</mo><mo>…</mo><mo>,</mo><mroot><mrow><msub><mi>α</mi><mi>r</mi></msub></mrow><mrow><msup><mi>ℓ</mi><mrow><msub><mi>n</mi><mi>r</mi></msub></mrow></msup></mrow></mroot></mrow></math>
<tex-math>\root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}}</tex-math>
</alternatives>
</inline-formula>
) over <italic>K</italic>(<italic>ζℓm</italic>) for all <italic>n</italic>1,...<italic>, nr, m</italic>. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.
</abstract>

Let K be a number field, and let ℓ be a prime number. Fix some elements α1,...,αr of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive integers with m ⩾ ni for every i. We show that there exist computable parametric formulas (involving only a finite case distinction) to express the degree of the Kummer extension K(ζℓm, 



α1ℓn1,…,αrℓnr
\root {{\ell ^{{n_1}}}} \of {{\alpha _1}} , \ldots ,\root {{\ell ^{{n_r}}}} \of {{\alpha _r}}


) over K(ζℓm) for all n1,..., nr, m. This is achieved with a new method with respect to a previous work, namely we determine explicit formulas for the divisibility parameters which come into play.

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Department of Mathematics, Faculty of Science, Technology and Medicine

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Divisibility Parameters and the Degree of Kummer Extensions of Number Fields"}

Let K be a number field, and let ℓ be a prime number. Fix some elements α1,...,αr of K× which generate a subgroup of K× of rank r. Let n1,...,nr, m be positive...

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Published Online: Feb 02, 2022

Page range: 71 - 88

Received: Oct 13, 2020

Accepted: Aug 30, 2021

DOI: https://doi.org/10.2478/udt-2021-0008

Keywords
number field, Kummer theory, Kummer extension, degree

© 2021 Antonella Perucca et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Divisibility Parameters and the Degree of Kummer Extensions of Number Fields

Published Online: Feb 02, 2022

Page range: 71 - 88

Received: Oct 13, 2020

Accepted: Aug 30, 2021

DOI: https://doi.org/10.2478/udt-2021-0008

Keywordsnumber field, Kummer theory, Kummer extension, degree

© 2021 Antonella Perucca et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
number field, Kummer theory, Kummer extension, degree