The Monogeneity of Kummer Extensions and Radical Extensions

2019/09/16 by Hanson Smith, Smith, Hanson · 1 citation
Mathematics · #11R04 #11R18 #11R20 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1909.07184

openalex publication_date 2019/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give necessary and sufficient conditions for the Kummer extension K:=ℚ(ζ_n,√[n]α) to be monogenic over ℚ(ζ_n) with √[n]α as a generator, i.e., for O_K=ℤ[ζ_n][√[n]α]. We generalize these ideas to radical extensions of an arbitrary number field L and provide necessary and sufficient conditions for √[n]α to generate a power O_L-basis for O_L(√[n]α). We also give sufficient conditions for K to be non-monogenic over 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 ϕ(n) to have ζ_n as a monogenic generator.

Citations

• Non monogénéité de l'anneau des entiers des extensions cycliques de Q de degré premier l ≥ 5
• A Dedekind's Criterion over Valued Fields

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• The minimal periodicity for integral bases of pure number fields

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