Leopoldt's conjecture for some Galois extensions (Q2906521)

This paper should be treated as a report (in a special case) of the author's attempt to prove Leopoldt's conjecture.NEWLINENEWLINELet \(K\) be a number field and let \(r_2\) denote the number of pairs of complex embeddings. Fix a prime \(p\). Then the number of independent \(\mathbb Z_p\)-extensions of \(K\) is \(1+r_2+\delta\) for some \(\delta \geq 0\). The integer \(\delta\) is called the Leopoldt defect and Leopoldt's conjecture asserts that \(\delta = 0\) (of course, there are many equivalent formulations). Leopoldt's conjecture is known if \(K\) is an abelian extension of \(\mathbb Q\) or of an imaginary quadratic number field as shown by \textit{A. Brumer} [Mathematika 14, 121--124 (1967; Zbl 0171.01105)]. There are very few further results.NEWLINENEWLINEIn the paper under review the author claims to show that if Leopoldt's conjecture holds for \(K\), then it holds for every solvable extension of \(K\). In particular, this would imply Leopoldt's conjecture for every solvable extension of \(\mathbb Q\). However, the proofs are only sketched and the reviewer has not been able to verify all the details.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00048].