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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].

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Linguistics knowledge representation and its relation to context abstraction are presented in brief. Nourani (e.g. Nourani 1999a) has put forth new visual computing techniques for intelligent multimedia context abstraction with linguistics components.In the present paper we also instantiate proof tree leaves with free Skolemized trees. Thus virtual trees, at times like intelligent trees, are substituted for the leaves. By a virtual tree we mean a term made up of constant symbols and named but not always prespecified Skolem function terms. In virtual planning with generic diagrams that part of the plan that involves free Skolemized trees is carried along with the proof tree for a plan goal. We can apply predictive model diagrams to compute queries and discover data knowledge from observed data and visual object images keyed with diagram functions. Model-based computing can be applied to automated data and knowledge engineering with keyed diagrams. Specific computations can be carried out with predictive model diagrams. For cognition, planning, and learning the robot's mind, a diagram grid can define state. The starting space applicable project was meant for an autonomous robots space journeys.