Primary components of the ideal class group of the \(\mathbb Z_p\)-extension over \(\mathbb Q\) for typical inert primes (Q816833)

Let \(p\) be an odd prime, \(\mathbb Z_p\) the ring of \(p\)-adic integers, and \(l\) a prime number different from \(p\). In [J. Lond. Math. Soc. (2) 66, No. 2, 257--275 (2002; Zbl 1011.11072)] the author has shown that, if \(l\) is a sufficiently large primitive root modulo \(p^2\), then the \(l\)-class group of the \(\mathbb Z_p\)-extension over the rational field is trivial. In the paper under review he modifies part of the proof of the above result and obtains, in the case \(p\leq 7\), that the result holds without assuming \(l\) to be sufficiently large.