I apologize in advance if this is easy, but I've tried Googling, and had no luck.
I'm currently working on a proof, and I realized in the course of writing that this proof will break if out there in the world there exists a division algebra $D$ with the following properties:
- The characteristic of $D$ (as a normal ring) is $p$.
- The degree of $D$ (the square root of its dimension over its center) is divisible by $p$.
I would really appreciate if anyone knows an example, or a proof that there are none. I could easily add hypotheses to rule out this case, but that would make things a bit messier, so I don't want to do that unless it's really necessary. Unfortunately, I know zippo about division algebras in characteristic p, other than that every finite one is actually just a field.
EDIT: Thanks for the references. Doing a little reading based on Mikhail's suggestion, I found a result which is good enough for my purposes: no such division algebra exists whose center is a perfect field, which was proven in 1934 by the remarkably named Abraham Adrian Albert. Those wanting more details on his life can read this detailed and impressive obituary by Jacobson. The best detail is that his Ukrainian father decided to ditch his last name (which is now unknown!) in Victorian England, and adopt the prince consort's name instead.