Families of extensions of \(\mathfrak l\)-rational number fields (Q679111)

Let \(\ell\) be a prime and let \(K\) be a number field containing the \(\ell\)th roots of unity (totally real, if \(\ell=2\)). Let \(\mathfrak l\) be a prime of \(K\) dividing \(\ell\). The field \(K\) is called \(\mathfrak l\)-rational if the maximal abelian \(\ell\)-extension of \(K\), which is unramified at finite primes not dividing \(\ell\) and in which \(\mathfrak l\) splits completely, is trivial. In an earlier paper [ibid. 6, 407-420 (1994; Zbl 0834.11049)], the author classified (for odd \(\ell\)) all cyclic \(\ell\)-extensions \(L/K\) such that \(L\) and \(K\) are both \(\ell\)-regular. In the present paper the same is done with ``\(\ell\)-regular'' replaced by ``\(\mathfrak l\)-rational'',\ resp.\ ``\(\mathfrak L\)-rational'',\ where \(\mathfrak L\) is a prime of \(L\) dividing \(\mathfrak l\). This is a generalization, because the concept ``\(\mathfrak l\)-rational'' is more general than \(\ell\)-regularity.