Throughout this post, is a field and is a finite dimensional central simple -algebra of degree
Let and suppose that is the characteristic polynomial of We know from linear algebra that and We also know that is -linear, is multiplicative and for all We’d like to extend the concepts of trace and determinant to any finite dimensional central simple algebra.
Definition. Let be the reduced characteristic polynomial of (see the definition of reduced characteristic polynomials in here). The reduced trace and the reduced norm of are defined, respectively, by and
Remark. Let be a splitting field of with a -algebra isomorphism So and because is the characteristic polynomial of
Theorem. 1) The map is -linear and the map is multiplicative.
2) for all
3) If then and
4) if and only if is a unit of So is a group homomorphism.
5) and are invariant under isomorphism of algebras and extension of scalars.
Proof. Fix a splitting field of and a -algebra isomorphism
1) We have already proved that the values of and are in Now, let and Then
2) This part is easy too:
3) Let be the identity element of Then
4) If for some then and thus Conversely, if then and so is invertible in Let be the inverse of Then for some because is surjective. Since is injective, it follows that Now if is not a unit of then it is a zero divisor because is artinian. So for some But then contradiction!
5) By Prp 3 and Prp 4 in this post, reduced characteristic polynomials are invariant under those things.