Every finite division ring has order $|Z|^n$ for some $n\in\Bbb N$ and where $Z$ is the center.

Question

I'm reading the proof of Ernst Witt that shows that every finite division ring is a field.

The proof begins with a statement that I don't fully understand:

Every finite division ring $R$ has order $|Z|^n$ for some $n\in\Bbb N$ and where $Z$ is the center of $R$.

So I guess that this can be shown by showing that $R$ can be written as a $n$ dimensional vector space over $Z$. Because such a vector space would have $|Z|^n$ elements.

I'm not sure how I can see that $R$ is such a vector space over $Z$. How can I show that?

Answer 1

If $Z$ is a field, then the division ring is a $Z$ vector space of dimension $n$ this implies that its cardinal is $|Z|^n$. Take a basis of the division ring to realize a bijection between $Z^n$ and the division ring.