## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 51, Number 2 (1999), 387-402.

### On quadratic extensions of number fields and Iwasawa invariants for basic ${Z}_{3}$-extensions

#### Abstract

Let ${Z}_{3}$ be the ring of 3-adic integers. For each number field $F$, let ${F}_{\infty ,3}$ denote the basic ${Z}_{3}$-extension over $F;$ let ${\lambda}_{3}\left(F\right)$ and ${\mu}_{3}\left(F\right)$ denote respectively the Iwasawa $\lambda \text{-}$ and $\mu $-invariants of ${F}_{\infty ,3}/F$. Here a number field means a finite extension over the rational field $Q$ contained in the complex field $C;F\subset C,$$$. Now let $k$ be a number field. Let $L$-denote the infinite set of totally imaginary quadratic extensions in $C$ over $k$(so that $L$-coincides with the set ${L}^{-}$ in the text when $k$ is totally real); let ${L}_{+}$ denote the infinite set of quadratic extensions in $C$ over $k$ in which every infinite place of $k$ splits (so that ${L}_{+}$ coincides with the set ${L}^{+}$ in the text when $k$ is totally real). After studying the distribution of certain quadratic extensions over $k$, that of certain cubic extensions over $k$, and the relation between the two distributions, this paper proves that, if $k$ is totally real, then a subset of $\{K\in {L}_{-}|{\lambda}_{3}\left(K\right)={\lambda}_{3}\left(k\right),{\mu}_{3}\left(K\right)={\mu}_{3}\left(k\right)\}$ has an explicit positive density in $L$-. The paper also proves that a subset of $\{L\in {L}_{+}|{\lambda}_{3}\left(L\right)=$${\mu}_{3}\left(L\right)=0\}$ has an explicit positive density in ${L}_{+}$ if 3 does not divide the class number of $k$ but is divided by only one prime ideal of $k$. Some consequences of the above results are added in the last part of the paper.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 51, Number 2 (1999), 387-402.

**Dates**

First available in Project Euclid: 10 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1213108023

**Digital Object Identifier**

doi:10.2969/jmsj/05120387

**Mathematical Reviews number (MathSciNet)**

MR1674755

**Zentralblatt MATH identifier**

0927.11052

**Subjects**

Primary: 11R23: Iwasawa theory

Secondary: 11R11: Quadratic extensions 11R29: Class numbers, class groups, discriminants 11R45: Density theorems

**Keywords**

Quadratic extension Iwasawa invariant basic $Z_{3}$-extension

#### Citation

HORIE, Kuniaki; KIMURA, Iwao. On quadratic extensions of number fields and Iwasawa invariants for basic $Z_{3}$ -extensions. J. Math. Soc. Japan 51 (1999), no. 2, 387--402. doi:10.2969/jmsj/05120387. https://projecteuclid.org/euclid.jmsj/1213108023