It is shown that the Steinitz invariants of the cubic extensions of a number field are uniformly distributed in the class group when the cubic extensions are ordered by the ideal norm of their relative discriminants. This remains true even if the extensions are restricted by specifying their splitting type at a finite number of places. The same statement is also proved for quadratic extensions.

We evaluate the character of the reflection representation of the split special orthogonal group over a finite field $F$ of odd characteristic. The value of the character at $g$ is expressed in terms of the dimensions of the eigenspaces of $g$ for eigenvalues in $F$ and in the norm 1 torus in the quadratic extension of $F$, and the behavior of the restriction of the quadratic form preserved by the group to the 1 and ${-}1$ eigenspaces of $g$.

We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

The space of integral 3-tensors is under the standard action of . A notion of primitivity is defined in this space and the number of primitive classes of a given discriminant is evaluated in terms of the class number of primitive binary quadratic forms of the same discriminant. Classes containing symmetric 3-tensors are also considered and their number is related to the 3-rank of the class group.