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EXPLICIT KUMMER GENERATORS FOR CYCLOTOMIC EXTENSIONS
Keywords:
Kummer theory, Kummer extension, number field, cyclotomic field, quadratic field, degreeDOI:
https://doi.org/10.17654/0972555522004Abstract
If $p$ is a prime number congruent to 1 modulo 3 , then we explicitly describe an element of the cyclotomic field $\mathbb{Q}\left(\zeta_3\right)$ whose third root generates the cubic subextension of $\mathbb{Q}\left(\zeta_{3 p}\right) / \mathbb{Q}\left(\zeta_3\right)$. Similarly, if $p$ is a prime number congruent to 1 modulo 4 , then we explicitly describe an element of the cyclotomic field $\mathbb{Q}\left(\zeta_4\right)$ whose fourth root generates the quartic cyclic subextension of $\mathbb{Q}\left(\zeta_{4 p}\right) / \mathbb{Q}\left(\zeta_4\right)$. For further number fields, we express generators of Kummer extensions inside cyclotomic fields in terms of Gauss sums.
Received: September 3, 2021
Revised: November 4, 2021
Accepted: November 18, 2021
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