## On Vandiver’s conjecture and $${\mathbb{Z}}_ p$$-extensions of $${\mathbb{Q}}(\zeta_{p^ n})$$.(English)Zbl 0709.11058

Let $$K_ n={\mathbb{Q}}[\zeta_{n+1}]$$, where $$\zeta_{n+1}$$ is a primitive $$p^{n+1}$$-th root of unity, p an odd prime, and let $$K_ n^+$$ be the maximal totally real subfield of K. Vandiver’s conjecture is that p does not divide $$h_ 0^+$$, the class number of $$K_ 0^+$$. Let $$R_ n={\mathcal O}_{K_ n}[1/p]$$; the authors relate Vandiver’s conjecture to the structure of $${\mathbb{Z}}_ p$$-Galois extensions of $$R_ 0$$, as follows. They first prove that if $$C_ n$$ is the p-Sylow subgroup of the ideal class group of $$R_ n$$, then the canonical map $$C_ n\to C_ m$$ is injective for any $$m>n$$ iff every $${\mathbb{Z}}_ p$$-extension of $$R_ n$$ has normal basis. They use this to show: Vandiver’s conjecture implies that each $${\mathbb{Z}}_ p$$-extension of $$R_ n$$ has normal basis for all n, and conversely provided that the Iwasawa invariant $$\lambda^+$$ of the cyclotomic $${\mathbb{Z}}_ p$$-extension of $$K_ 0^+$$ is zero. Finally they prove that Vandiver’s conjecture is equivalent to the property that each cyclic extension of $$K_ 0$$ lying in $$H^-$$ is contained in a $${\mathbb{Z}}_ p$$-extension of $$K_ 0$$, where H is the Hilbert class field of $$K_ 0$$ with Galois group $$G=G^+\oplus G^-$$ and $$H^- =H^{G^+}$$. These results strengthen the plausibility of Vandiver’s conjecture because it is known from other work of the authors [J. Number Theory 32, 131-150 (1989; cf. the preceding review)] that in the group of $${\mathbb{Z}}_ p$$-extensions of $$R_ n$$, the group of $${\mathbb{Z}}_ p$$- extensions with normal basis has finite index.
Reviewer: L.N.Childs

### MSC:

 11R18 Cyclotomic extensions 11R23 Iwasawa theory

Zbl 0709.11057
Full Text:

### References:

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