LAUSR

In this paper, we investigate the cyclicity of the [Formula: see text]-class group of the first Hilbert [Formula: see text]-class field of some quadratic number field whose discriminant is not a sum of two squares. For this, let [Formula: see text] be different prime integers. Put [Formula: see text], and denote by [Formula: see text] its [Formula: see text]-class group and by [Formula: see text] (respectively [Formula: see text]) its first (respectively second) Hilbert [Formula: see text]-class field. Then, we are interested in studying the metacyclicity of [Formula: see text] and the cyclicity of [Formula: see text] whenever the [Formula: see text]-rank of [Formula: see text] is [Formula: see text].