Abstract
Let K be a number field, and let α1,.,αr be elements of K× which generate a subgroup of K× of rank r. Consider the cyclotomic-Kummer extensions of K given by K(ζn,α1n1,.,αrnr), where ni divides n for all i. There is an integer x such that these extensions have maximal degree over K(ζg,α1g1,.,αrgr), where g =gcd(n,x) and gi =gcd(ni,x). We prove that the constant x is computable. This result reduces to finitely many cases the computation of the degrees of the extensions K(ζn,α1n1,.,αrnr) over K.
| Original language | English |
|---|---|
| Pages (from-to) | 1091-1110 |
| Number of pages | 20 |
| Journal | International Journal of Number Theory |
| Volume | 17 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - Jun 2021 |
| Externally published | Yes |
Keywords
- Kummer extension
- Kummer theory
- Number field
- degree
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Perucca, A., Sgobba, P., & Tronto, S. (2021). The degree of Kummer extensions of number fields. International Journal of Number Theory, 17(5), 1091-1110. https://doi.org/10.1142/S1793042121500263