**Theorem**. Let be a finite dimensional central division -algebra of degree and let Suppose that is the minimal polynomial of over Then

*Proof*. We have and hence Now, the set is a -basis for Let be a -basis for Then the set

is a -basis for Now define by for all Let be the characteristic polynomial of By the example in this post, we have

We are now going to find in terms of To do so, we first find the matrix of with respect to the ordered basis Notice that since we have

Now, since for all and we have the following block matrix

where It is easy to see that

Thus gives us

Hence by and That means is the unique irreducible factor of and hence for some integer Now, since and we must have

**Example**. (You should also see the example in this post!) Let be the division algebra of real quaternions and It is not hard to see that is the minimal polynomial of over Therefore, since we get from the above theorem that