Scaled trace forms of central simple algebras (Q1815269)

Let \(A\) be a central simple algebra over a field \(F\) of characteristic not two. Put \(n^2=\dim_F A\). Let \(z\) be a non-zero element of \(A\). The mapping \(q_z:A\to F\) given by \(q_z(x)= tr(zx^2)\) is called a scaled trace form. For \(z=1\) one obtains the ordinary trace forms. The paper examines scaled trace forms \(q_z\) from the viewpoint of the algebraic theory of quadratic forms. In section 1 the basic definitions and necessary and sufficient conditions for the scaled trace forms to be non-singular are given. In section 2 the case where \(A\cong M_n(F)\) is dealt with. In section 3 the algebraic invariants of scaled trace forms are studied. It is shown that \(\text{det }q_z\) is up to sign equal to the reduced norm \(nr(z)\). A formula for the signatures is given in case \(A\) is a quaternion or a biquaternion algebra. Sections 4 and 5 relate scaled trace forms to the transfer homomorphism. This relation is used to obtain information about the kernel and the image of the extension of scalar homomorphisms of Witt rings.