Algebraic number fields with the discriminant equal to that of a quadratic number field. (English) Zbl 0865.11074

Kondo, Takeshi

J. Math. Soc. Japan 47, No. 1, 31-36 (1995).

Let \(d(F)\) be the discriminant of an algebraic number field \(F\) of degree \(n\). In this paper, the author intends to generalize some results on unramified Galois extensions obtained by several authors under the assumption that \(d(F)\) is square free [cf. K. Yamamura, J. Fac. Sci., Univ. Tokyo, Sect. I A 38, 99-135 (1991; Zbl 0747.14001)]. Namely, he proves that if \(d(F)\) is equal to the discriminant of a quadratic number field (i.e. \(d(F)\) is not a square and is equal to the discriminant of the field \(\mathbb{Q}(\sqrt{d(F)}))\), then the Galois group of the Galois closure \(K\) of \(F\) over the rational number field \(\mathbb{Q}\) is isomorphic to the symmetric group of degree \(n\) and the extension \(K/\mathbb{Q}(\sqrt {d(F)})\) is unramified at all finite primes of \(\mathbb{Q}(\sqrt{d(F)})\). Moreover, he provides a necessary and sufficient condition for \(d(F)\) to be equal to the discriminant of \(\mathbb{Q}(\sqrt{d(F)})\) in terms of the ramification index and the residue class degree.

Reviewer: H.Yokoi (Iwasaki)

MSC:

11R32	Galois theory
11R29	Class numbers, class groups, discriminants
11R11	Quadratic extensions

Keywords:

discriminant; unramified Galois extensions; quadratic number field; Galois group; symmetric group

Citations:

Zbl 0747.14001