Quaternion Orders and Ternary Quadratic Forms Orders of Class Number One and Representations of Algebraic Integers by Quadratic Forms
Doktorsavhandling, 2000

Let R be the ring of integers in a totally real quadratic field K. The purpose of the thesis is to study totally definite quaternion R-orders and representations of elements in R by totally positive definite integral ternary quadratic forms. The thesis consists of three papers. In the first paper, we prove that R-orders of class number one in totally definite quaternion algebras over K only exist for K=Q(Sqrt(d)) with d=2,5,13,17. We show that there are twenty-eight isomorphism classes of such orders for d=2, twenty-five for d=5, nine for d=13 and thirteen for d=17. We describe one order L from each of these classes by finding a ternary quadratic form f such that the even part of the Clifford algebra C0(f) is isomorphic to L. In the second paper, we study representations of totally positive numbers N .epsilon. R, where R is a principal ideal domain in a totally real number field K, by totally positive definite ternary quadratic forms over R. We prove a quantitative formula relating the number of representations of N by different classes in the genus of a form f to the class number of R[Sqrt({-cfN})], where cf .epsilon. R is a constant only depending on f. We give an algebraic proof of a classical result of H. Maass on representations by sums of three squares over the integers R=Z[(1+Sqrt(5))/2]. We also obtain an explicit formula for the number of primitive representations of N .epsilon. R by the sum of three squares, relating the number of representations to the class number of R[Sqrt({-N})]. In the third paper, we study simultaneous embeddings of two maximal orders in totally imaginary quadratic extensions of K=Q(Sqrt(d)), for d=2,5,13,17, into totally definite quaternion algebras A over K. We give necessary and sufficient conditions under which the images of two embeddings generate an R-order in A. We find class numbers relations between the embedded orders and also some applications to representations of totally positive numbers N .epsilon. R by certain quadratic forms.

even Clifford algebra


class number


quaternion order

ternary quadratic form




embedding number

quaternion algebra


Elise Björkholdt

Institutionen för matematik

Göteborgs universitet





Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 1607