Unramified Quaternion Extensions of Quadratic Number Fields

The first mathematician who studied quaternion extensions (H8-extensions for short) was Dedekind [6]; he gave Q( √ (2 + √ 2)(3 + √ 6) ) as an example. The question whether given quadratic or biquadratic number fields can be embedded in a quaternion extension was extensively studied by Rosenblüth [32], Reichardt [31], Witt [36], and Damey and Martinet [5]; see Ledet [19] and the surveys [15] and [10] for more details. Later, Fujisaki [8], Kiming [16] and Vaughan [35] gave simple constructions of H8-extensions of Q. In [1], Bachoc and Kwon studied H8-extensions of cyclic cubic number fields from an arithmetic viewpoint. The corresponding problem for certain sextic fields was dealt with by Jehanne [14] and Cassou-Noguès and Jehanne [2]. Quaternion extensions of Q also played a central role in the theory of the Galois module structure of the ring of integers of algebraic number fields (see Martinet’s papers [26, 27]), and the Introduction of [7] for a detailed account). As Cohn [4] showed, quaternion extensions can also be used to explain congruences between certain binary quadratic forms. Since quaternion extensions of Q always ramify over their biquadratic subfield (see Cor. 2 below), they do not occur as Hilbert class fields of quadratic or biquadratic number fields. In order to find unramified H8-extensions one has to look at base fields 6= Q. Already Furtwängler [9] knew that such extensions exist, Koch [18] gave explicit criteria that guarantee their existence, and it was Kisilevsky [17] who showed that the second Hilbert 2-class field of e.g. k = Q( √ −30 ) is an H8-extension of k. Hettkamp [12] found criteria for the existence of unramified H8extensions of certain real quadratic number fields, and finally M. Horie [13] gave the first explicit example of such an extension. Recently, Louboutin and Okazaki [22, 23, 24] have computed relative class numbers of quaternion CM-extensions of Q as well as of unramified quaternion extensions of real quadratic number fields ([25]). In [20, 21] we have shown how to construct unramified quaternion extensions of a number field k which is a quadratic extension of a field F , and F is totally real and has odd class number in the strict sense. In this article we will show that this construction can be carried out with completely elementary methods (along the same lines as the classical papers of Rédei [29] and Rédei & Reichardt [30]) as long as we restrict ourselves to quadratic number fields.