A field of characteristic 0 is \(\tilde p\)-closed if it contains a primitive p-th root of unity and does not have any proper p-extension. Using results of J. Neukirch [Invent. Math. 6, 296-314 (1969; Zbl 0192.401) and J. Reine Angew. Math. 238, 135-147 (1969; Zbl 0201.059)] the author obtains the following theorem 1: Let p be a prime number and let \(\Omega/K\) be a \(\tilde p\)-closed Galois extension of algebraic number fields. Then the following are equivalent: (a) There exists a non- archimedean valuation \(v| p\) of K such that K is \(\Omega\)-henselian with respect to v and \([K_ v:{\mathbb{Q}}_ p]<\infty\). (b) There exists a finite extension \(K'/{\mathbb{Q}}_ p\) and a \(\tilde p\)-closed Galois extension \(\Omega'/K'\) such that Gal\((\Omega/K)\cong Gal(\Omega'/K')\). Furthermore, v is unique and one has \([K_ v:{\mathbb{Q}}_ p]=[K':{\mathbb{Q}}_ p].\)
Also, a result of K. Uchida [Tôhoku Math. J., II. Ser. 31, 359- 362 (1979; Zbl 0422.12006)] is shown to hold for Galois extensions which are \(\tilde p\)-closed for all primes p in a set of Dirichlet density 1.