Describe , where denotes the Hamiltonian Quaternions. Prove that is a subring of which is a field but is not contained in .

We claim that . The direction is clear. To see the direction, let and .

Using the distributive property and the fact that multiplication of real numbers is commutative, we see that , while .

Since is in the center, these elements are equal, and thus their corresponding coefficients are equal. This yields three equations:

where we consider , , and to be fixed while , , and are arbitrary real numbers. Consider the first equation; we may assume that . Then, if , we have . This is a contradiction since and are fixed while is arbitrary. Thus . Similarly, , and we have .

Now let . Let and . Since and , by the subgroup criterion . Since , is a subring of .

Now if , we see that ; thus has an inverse in , specifically . Thus is a division ring. Moreover, , so that is a field.

Note, however, that while , so that .