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


11R18 Cyclotomic extensions
11R23 Iwasawa theory


Zbl 0709.11057
Full Text: DOI


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