Using Cohen's construction of defining polynomials for a cyclic group of odd prime order, we define a polynomial with some parameters which generates cyclic extensions of a given odd prime degree, and prove it to be generic in the sense as defined below.

In this paper, we prove by using Ichimura-Sumida criterion the vanishing of the Iwasawa $\lambda_3$-invariant of the real quadratic field $\mathbf{Q}(\sqrt{39345017} )$. We use also the fact that the field has the infinite 3-class field tower.

We prove an analogue of Minkowski's second fundamental theorem for a vector space over a central division algebra in an adelic manner.

In this paper we give characterizations of quaternionic space forms in the class of quaternionic Kähler manifolds in terms of the curvature tensor and the extrinsic shape of geodesics on geodesic spheres.