## Journal of the Mathematical Society of Japan

### 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 $\lambda_{3}(F)$ and $\mu_{3}(F)$ denote respectively the Iwasawa $\lambda-$ 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,$$[F:Q]<\infty$. Now let $k$ be a number field. Let $L$-denote the infinite set of totally imaginary quadratic extensions in $C$ over $k$(so that $L$-coincides with the set $\mathit{L}^{-}$ in the text when $k$ is totally real); let $\mathit{L}_{+}$ denote the infinite set of quadratic extensions in $C$ over $k$ in which every infinite place of $k$ splits (so that $\mathit{L}_{+}$ coincides with the set $\mathit{L}^{+}$ in the text when $k$ is totally real). After studying the distribution of certain quadratic extensions over $k$, that of certain cubic extensions over $k$, and the relation between the two distributions, this paper proves that, if $k$ is totally real, then a subset of $\{K\in \mathit{9}_{-}|\lambda_{3}(K)=\lambda_{3}(k),\mu_{3}(K)=\mu_{3}(k)\}$ has an explicit positive density in $L$-. The paper also proves that a subset of $\{L\in \mathit{9}_{+}|\lambda_{3}(L)=$$\mu_{3}(L)=0\}$ has an explicit positive density in $\mathit{L}_{+}$ 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

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

Digital Object Identifier
doi:10.2969/jmsj/05120387

Mathematical Reviews number (MathSciNet)
MR1674755

Zentralblatt MATH identifier
0927.11052

#### 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