Unramified Quadratic Extensions of a Quadratic Field

Unramified Alternating Extensions of Quadratic Fields

We exhibit, for each n ≥ 5, infinitely many quadratic number fields admitting unramified degree n extensions with prescribed signature whose normal closures have Galois group An. This generalizes a result of Uchida and Yamamoto, which did not include the ability to restrict the signature, and a result of Yamamura, which was the case n = 5. It is a folk conjecture that for n ≥ 5, all but finitel...

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...

Elliptic units of cyclic unramified extensions of complex quadratic fields

Spreads of Right Quadratic Skew Field Extensions

~ ABSTRACT. Let L/K be a right quadratic (skew) field extension and let P be a 3-dimensional projective space over K which is embedded in a 3dimensional projective space P over L. Moreover let I be a line of P which ~ carries no point of P. The main result is that even when L or K is a skew ~ field the following holds true: A desarguesian spread of P is given by the ~ ~ set of all lines of P wh...

Hecke’s integral formula for quadratic extensions of a number field

Let K/F be a quadratic extension of number fields. After developing a theory of the Eisenstein series over F , we prove a formula which expresses a partial zeta function of K as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker’s type which relates the 0-th Laurent coefficients at s = 1 of zeta functions of K and F .