It is shown that an even partition A∪B of the set R = {1, 2, . . . , p− 1} of positive residues modulo an odd prime p is the partition into quadratic residues and quadratic non-residues if and only if the elements of A and B satisfy certain additive properties, thus providing a purely additive characterization of the set of quadratic residues. 1 Additive properties of quadratic residues An inte...

We study the index of the group of units in the genus field of an imaginary quadratic number field modulo the subgroup generated by the units of the quadratic subfields (over ‫)ޑ‬ of the genus field.

The modern theory of class field towers has its origins in the study of the p-class field tower over a quadratic imaginary number field, so it is fitting that this problem be the first in the discipline to be nearing a solution. We survey the state of the subject and present a new cohomological condition for a quadratic imaginary number field to have an infinite p-class field tower (for p odd)....

We present recent results on the computation of quadratic function fields with high 3-rank. Using a generalization of a method of Belabas on cubic field tabulation and a theorem of Hasse, we compute quadratic function fields with 3-rank ≥ 1, of imaginary or unusual discriminant D, for a fixed |D| = q. We present numerical data for quadratic function fields over F5, F7, F11 and F13 with deg(D) ≤...