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The monogeneity of radical extensions

Volume 198 / 2021

Hanson Smith Acta Arithmetica 198 (2021), 313-327 MSC: 11R04, 11R18, 11R20. DOI: 10.4064/aa200811-7-10 Published online: 22 February 2021

Abstract

Let $L$ be a number field. We give necessary and sufficient conditions for a radical extension $L(\!\sqrt [n]{\alpha })$ to be monogenic over $L$ with $\sqrt [n]{\alpha }$ as a generator, i.e., for $\sqrt [n]{\alpha }$ to generate a power $\mathcal {O}_L$-basis for the ring of integers $\mathcal {O}_{L(\!\sqrt [n]{\alpha })}$. We also give sufficient conditions for a Kummer extension of the form $\mathbb {Q}(\zeta _n,\sqrt [n]{\alpha })$ to be non-monogenic over $\mathbb {Q}$ and establish a general criterion relating ramification and relative monogeneity. Using this criterion, we find a necessary and sufficient condition for a relative cyclotomic extension of degree $\phi (n)$ to have $\zeta _n$ as a monogenic generator.

Authors

  • Hanson SmithDepartment of Mathematics
    University of Connecticut
    341 Mansfield Road U1009
    Storrs, CT 06269-1009, U.S.A.
    e-mail

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