Here you can see part (1).
Introduction. Let’s take a look at a couple of properties of the trace map in matrix rings. Let be a field and Let be the standard basis for Let where Now, where is the identity element of Now let’s define See that and for all We are going to extend these facts to any finite dimensional central simple algebra.
Notation. I will assume that is a finite dimensional central simple -algebra.
Theorem. There exists a unique element such that for all Moreover, and for all
Proof. As we saw in this theorem, the map defined by is a -algebra isomorphism. Let’s forget about the ring structure of for now and look at it just as a -vector space. Then and so we have a -vector space isomorphism defined by Since there exists a unique element
such that Then
To prove we choose a splitting field of of Then for some integer which is the degree of Let’s identify and with and respectively. Then and become subalgebras of and Let and Recall that, by the last part of the theorem in part (1), for all Let and Then, since is the center of we have
Thus by the uniqueness of Hence But is a matrix ring and so, as we mentioned in the introduction, So
To prove the last part of the theorem, let Then
The last equality holds by the second part of the theorem in part (1). Also, the image of is in the center of and so commutes with Now, we also have
Thus and so
Definition. The element in the theorem is called the Goldman element for