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April 2014 Notes on the existence of unramified non-abelian $p$-extensions over cyclic fields
Akito Nomura
Proc. Japan Acad. Ser. A Math. Sci. 90(4): 67-70 (April 2014). DOI: 10.3792/pjaa.90.67

Abstract

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.

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Akito Nomura. "Notes on the existence of unramified non-abelian $p$-extensions over cyclic fields." Proc. Japan Acad. Ser. A Math. Sci. 90 (4) 67 - 70, April 2014. https://doi.org/10.3792/pjaa.90.67

Information

Published: April 2014
First available in Project Euclid: 1 April 2014

zbMATH: 1343.12004
MathSciNet: MR3189510
Digital Object Identifier: 10.3792/pjaa.90.67

Subjects:
Primary: 12F12
Secondary: 11R16 , 11R29

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

Rights: Copyright © 2014 The Japan Academy

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Vol.90 • No. 4 • April 2014
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