On Vandiver's conjecture and Zp-extensions of Q(ζpn)

AbstractSuppose K+ is the maximal totally real subfield of K = Q(ζp) and h+ is the class number of K+. Vandiver's conjecture states that p does no divide h+. Each Zp-extension extension K∞ of K defines a unique Zp-extension R∞ of R = x[p−1] with k being the ring of integers of K. In this paper we prove for all odd primes p, (1) if p does not divide h+, then each Zp-extension of R has a normal basis over R. Conversely, if each Zp-extension of R has a normal basis over R and if the Iwasawa invariant λ+ of the cyclotomic Zp-extension of K+ is zero, then p does not divide h+. (2) p does not divide h+ if and only if each cyclic extension of K, lying in the minus part H− of the Hilbert p-class field H of K, is contained in a Zp-extension of K