## Capitulation in the cyclotomic $$\mathbb{Z}_2$$ extension of CM number fields.(English)Zbl 1462.11099

Summary: Let $$\mathbb{K}_n$$ be the intermediate steps in the cyclotomic $$\mathbb{Z}_p$$-extension of a CM number field $$\mathbb{K}$$. For $$p\neq 2$$ the minus part of the $$p$$-class groups is given by $$A_n^-=\frac{1}{2}(1-j)A_n$$. We will give a new definition of the minus part for $$p=2$$ and prove that there is no finite submodule in $$\lim_{\infty\leftarrow n}A_n^-$$. Furthermore we show that $$\mu=0$$ if and only if $$\mu^-=0$$ in this new definition.

### MSC:

 11R23 Iwasawa theory 11R18 Cyclotomic extensions
Full Text:

### References:

 [1] Washington, L. C., Introduction to Cyclotomic Fields, (1982), Springer · Zbl 0484.12001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.