The Kerr-Taub-NUT metric is a local analytic solution of the vacuum Einstein-Maxwell equations. When the metric is expressed in Schwarzschild-like coordinates, two types of coordinate singularity are present. One occurs at certain values of the ``radial'' coordinate where grr becomes infinite and corresponds to bifurcate Killing horizons; the other occurs at θ=0,π, where the determinant of the components of the metric vanishes. It is shown that for nonzero NUT parameter the fixed points of the bifurcate Killing horizons and the degeneracies at θ=0,π cannot all be covered in one manifold. A maximal analytic manifold is constructed which covers the degeneracies at θ=0,π. It is non-Hausdorff but contains maximal Hausdorff subspaces, topologically S3×R, which reduce to Taub-NUT space for vanishing Kerr parameter. Kerr-Taub space can be interpreted as a closed, inhomogeneous electromagnetic-gravitational wave undergoing gravitational collapse. Another maximal analytic manifold is constructed which covers the fixed points of the bifurcate Killing horizons and the degeneracy at θ=0. It is suggested that this manifold represents the superposition of the Kerr geometry and a massless source of angular momentum at θ=π characterized by the NUT parameter.

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The determination of the class number of totally real fields of large discriminant is known to be a difficult problem. The Minkowski bound is too large to be useful, and the root discriminant of the field can be too large to be treated by Odlyzko's discriminant bounds. We describe a new technique for determining the class number of such fields, allowing us to attack the class number problem for a large class of number fields not treatable by previously known methods. We give an application to Weber's class number problem, which is the conjecture that all real cyclotomic fields of power of 2 conductor have class number 1.