## Illinois Journal of Mathematics

### Notes on the existence of certain unramified 2-extensions

Akito Nomura

#### Abstract

We study the inverse Galois problem with restricted ramification. Let $K$ be an algebraic number field and $G$ be a $2$-group. We consider the question whether there exists an unramified Galois extension $M/K$ with Galois group isomorphic to $G$. We study this question using the theory of embedding problems. Let $L/k$ be a Galois extension and $(\varepsilon): 1\to \mathbf{Z}/2\mathbf{Z}\to E\to \operatorname{Gal} (L/k)\to 1$ a central extension. We first investigate the existence of a Galois extension $M/L/k$ such that the Galois group $\operatorname{Gal} (M/k)$ is isomorphic to $E$ and any finite prime is unramified in $M/L$. As an application, we prove the existence of an unramified extension over cyclic quintic fields with Galois group isomorphic to $32{\Gamma}_5a_2$ under the condition that the class number is even. We also consider the Fontaine-Mazur-Boston Conjecture in the case of abelian $l$-extensions over $\mathbf{Q}$.

#### Article information

Source
Illinois J. Math., Volume 46, Number 4 (2002), 1279-1286.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138479

Digital Object Identifier
doi:10.1215/ijm/1258138479

Mathematical Reviews number (MathSciNet)
MR1988263

Zentralblatt MATH identifier
1024.12005

Subjects
Primary: 12F12: Inverse Galois theory
Secondary: 11R29: Class numbers, class groups, discriminants 11R32: Galois theory

#### Citation

Nomura, Akito. Notes on the existence of certain unramified 2-extensions. Illinois J. Math. 46 (2002), no. 4, 1279--1286. doi:10.1215/ijm/1258138479. https://projecteuclid.org/euclid.ijm/1258138479