Jurgen Hurrelbrink

Annihilating polynomials for group rings and Witt rings

Natural annihilating polynomials for group rings are produced; this yields, as a special case, the annihilating polynomials for Witt rings that have been discovered only recently. 0. Introduction. For a positive interger n let the polynomial Ln G Z[t] be given by Ln(t) = (t n)(t n + 2) • . . . • (t + n 2)0 + n). Only recently it has been discovered that, for any field F of characteristic ^ 2, every n-dimensional quadratic form q over F satisfies

ON IDEAL CLASS GROUPS AND UNITS IN TERMS OF THE QUADRATIC FORM x2+32y2

For quadratic number fields with primes pj≡1 mod 8, the authors study the class number and the norm of the fundamental unit of F. The results generalize nicely what has been familiar for the fields with a prime p≡1 mod 8, including density statements. And the results are stated in terms of the quadratic form x2+32y2 and illustrated in terms of graphs.

Class Numbers, Units and K2

This paper deals with topics from Algebraic Number Theory and Algebraic K-Theory. It establishes relationships between the class number and the signs of units of an algebraic number field on one side, and the order and the structure of the 2-primary subgroup of the Milnor K-group over the ring of integers of the number field on the other side.

SPLITTING PATTERNS AND TRACE FORMS

The splitting pattern of a quadratic form q over a fieldk consists of all distinct Witt indices that occur for q over extension fields of k. In small dimensions, the complete list of splitting patterns of quadratic forms i s known. We show that all splitting patterns of quadratic forms of dimension at most n ine can be realized by trace forms.