Proceedings of the Japan Academy, Series A, Mathematical Sciences

Notes on the existence of unramified non-abelian $p$-extensions over cyclic fields

Akito Nomura

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We study the inverse Galois problem with restricted ramifications. Let $p$ and $q$ be distinct odd primes such that $p\equiv 1 \bmod q$. Let $E(p^{3})$ be the non-abelian group of order $p^{3}$ such that the exponent is equal to $p$, and let $k$ be a cyclic extension over $\mathbf{Q}$ of degree $q$. In this paper, we study the existence of unramified extensions over $k$ with the Galois group $E(p^{3})$. We also give some numerical examples computed with PARI.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 4 (2014), 67-70.

First available in Project Euclid: 1 April 2014

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Zentralblatt MATH identifier

Primary: 12F12: Inverse Galois theory
Secondary: 11R16: Cubic and quartic extensions 11R29: Class numbers, class groups, discriminants

Unramified $p$-extension inverse Galois problem ideal class group cyclic cubic field


Nomura, Akito. Notes on the existence of unramified non-abelian $p$-extensions over cyclic fields. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 4, 67--70. doi:10.3792/pjaa.90.67.

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