On unramified Galois extensions of real quadratic number fields
Let \(A_n\) be the alternating group of degree \(n\). By an \(A_n\)-extension we mean a Galois extension with Galois group \(A_n\). It is known that there exist infinitely many real (resp. imaginary) quadratic number fields having an \(A_n\)-extension which is unramified at all finite primes (resp. all primes) (cf. \textit{Y. Yamamoto} [ibid. 7, 57--76 (1970; Zbl 0222.12003)]). In this note, the author shows that there exist infinitely many real quadratic number fields \(F\) having an \(A_5\)-extension which is unramified at all primes including the infinite primes (''strictly unramified''). In order to prove this, he determines a certain family of fifth degree polynomials \(f(x)\) with discriminant \(D\) such that the splitting field \(K\) of \(f(x)\) is a strictly unramified \(A_5\)-extension over the real field \(F=\mathbb Q(\sqrt{D})\). In the last section, the author tackles the same problem for \(n\) larger than \(5\). A related work in this direction is due to \textit{J. Elstrodt, F. Grunewald} and \textit{J. Mennicke} [Glasg. Math. J. 27, 31--37 (1985; Zbl 0579.12005)].
scientific article; zbMATH DE number 3985353