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

<p>Let <italic>K</italic> be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of <italic>K</italic>, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of <italic>K</italic>. For <italic>G</italic> a finitely generated subgroup of <italic>K</italic><sup>×</sup>, we consider the cyclotomic-Kummer extensions <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0017_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>K</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub><mo>,</mo><mroot><mi>G</mi><mi>n</mi></mroot></mrow><mo>)</mo></mrow><mo>/</mo><mi>K</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right)</tex-math>
</alternatives>
</inline-formula> for all positive integers <italic>n</italic> and <italic>t</italic>, and we describe an explicit finite procedure to compute at once the degree of all these extensions.</p>
</abstract>

Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K×, we consider the cyclotomic-Kummer extensions 


K(ζnt,Gn)/K(ζnt)
K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right)

 for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Uniform distribution theory

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

{"article-title":"Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields"}

Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the...

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Published Online: Dec 12, 2022

Page range: 165 - 194

Received: Aug 31, 2021

Accepted: Oct 06, 2022

DOI: https://doi.org/10.2478/udt-2022-0017

Keywords
Kummer theory, Kummer extension, cyclotomic field, multiquadratic number field, quartic cyclic number field, cubic number field

© 2022 Flavio Perissinotto et al., published by Sciendo

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

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

Published Online: Dec 12, 2022

Page range: 165 - 194

Received: Aug 31, 2021

Accepted: Oct 06, 2022

DOI: https://doi.org/10.2478/udt-2022-0017

KeywordsKummer theory, Kummer extension, cyclotomic field, multiquadratic number field, quartic cyclic number field, cubic number field

© 2022 Flavio Perissinotto et al., published by Sciendo

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

Keywords
Kummer theory, Kummer extension, cyclotomic field, multiquadratic number field, quartic cyclic number field, cubic number field