## On quadratic extensions of number fields and Iwasawa invariants for basic $$\mathbb{Z}_3$$-extensions.(English)Zbl 0927.11052

Let $$\mathbb{Z}_3$$ be the ring of 3-adic integers. Assume that a number field, namely a finite extension over the rational field, is always contained in the complex field. For each number field $$F$$, let $$F_{\infty,3}$$ denote the basic $$\mathbb{Z}_3$$-extension over $$F$$, $$\lambda_3(F)$$ the Iwasawa $$\lambda$$-invariant of $$F_{\infty,3}$$, $$\mu_3(F)$$ the Iwasawa $$\mu$$-invariant of $$F_{\infty,3}/F$$. Given any number field $$k$$, let $$Q_-$$ denote the infinite set of totally imaginary quadratic extensions over $$k$$, and $$Q_+$$ the infinite set of quadratic extensions over $$k$$ in which every infinite place of $$k$$ splits. After studying the distribution of certain quadratic extensions over $$k$$, that of certain cubic extensions over $$k$$, and the relation between the two distributions, in this paper it is mainly proved that, if $$k$$ is totally real, then a subset of $$\{K\in Q_-\mid \lambda_3(K)= \lambda_3(k)$$, $$\mu_3(K)= \mu_3(k)\}$$ has an explicit positive density in $$Q_-$$.
The author also proves that the subset of $$\{L\in Q_+\mid \lambda_3(L)= \mu_3(L)= 0\}$$ has an explicit positive density in $$Q_+$$ 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.

### MSC:

 11R23 Iwasawa theory 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants
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