Abstract
Let \((K_{n})_{n \ge 1}\) be a tower of number fields whose defining polynomials are iterates of a polynomial f. We show that the sequence of class numbers \((h(K_{n}))_{n \ge 1}\) satisfies \(h(K_n) \mid h(K_{n+1})\) when f is a monic Eisenstein polynomial. When \(f(x)=x^2-c\) is a quadratic polynomial, we also determine when the ring of integers \({\mathcal {O}}_{K_{n}}\) equals \({\mathbb {Z}}[a_{n}]\), where \(a_{n}\) is a root of \(f^n\).
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Acknowledgements
This paper is based on Chapter 5 of the author’s PhD thesis [4]. The author would like to thank his advisor Su-ion Ih for his guidance and many helpful comments. The author also thanks the anonymous referees for many valuable suggestions.
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Li, R. On number fields towers defined by iteration of polynomials. Arch. Math. 119, 371–379 (2022). https://doi.org/10.1007/s00013-022-01770-w
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DOI: https://doi.org/10.1007/s00013-022-01770-w