×

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

Citations:

Zbl 0709.11057
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Chase, S.U.; Harrison, D.K.; Rosenberg, A., Galois theory and Galois cohomology of commutative rings, Mem. amer. math. soc., 52, 15-33, (1965) · Zbl 0143.05902
[2] Childs, L., Abelian Galois extensions of rings containing roots of unity, Illinois J. math., 15, 273-279, (1971) · Zbl 0211.37102
[3] Garfinkel, G.; Orzech, M., Galois extensions as modules over the group ring, Canad. J. math., 22, 242-248, (1970) · Zbl 0197.03401
[4] Greenberg, R., On the Iwasawa invariants of totally real number fields, Amer. J. math., 98, 263-284, (1976) · Zbl 0334.12013
[5] Harrison, D.K., Abelian extensions of commutative rings, Mem. amer. math. soc., 52, 1-14, (1965) · Zbl 0143.06003
[6] Iwasawa, K., On \(Z\)_{i}-extensions of algebraic number fields, Ann. of math., 98, 246-326, (1973) · Zbl 0285.12008
[7] Kersten, I.; Michaliček, J., On γ-extensions of totally real and complex multiplication fields, C. R. math. rep. sci. canad., IX, 309-314, (1987) · Zbl 0649.12010
[8] Kersten, I.; Michaliček, J., Kummer theory without roots of unity, J. pure appl. algebra, 50, 21-72, (1988) · Zbl 0638.13007
[9] Kersten, I.; Michaliček, J., \(Z\)_{p}-extensions of complex multiplication fields, J. number theory, 32, (1989) · Zbl 0709.11057
[10] Lang, S., ()
[11] Washington, L., ()
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.