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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:

  1. The characteristic of $D$ (as a normal ring) is $p$.
  2. 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.

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  • $\begingroup$ I believe that there exists such a division algebra. Probably, an example was constructed in Hochschild, G. Simple algebras with purely inseparable splitting fields of exponent 1. Trans. Amer. Math. Soc. 79, (1955). 477–489. $\endgroup$
    – Mikhail Bondarko
    Sep 11, 2014 at 17:14
  • $\begingroup$ Also, look at: Aljadeff, Eli(IL-TECH); Sonn, Jack(IL-TECH) Relative Brauer groups and m-torsion. (English summary) Proc. Amer. Math. Soc. 130 (2002), no. 5, 1333–1337. A related result that seems to be true: for a local field of characteristic $p$ the $p$-torsion of the Brauer group is non-zero (and is ind-cyclic). $\endgroup$
    – Mikhail Bondarko
    Sep 11, 2014 at 17:26
  • $\begingroup$ msp.org/pjm/1997/181-3/pjm-v181-n3-p07-s.pdf mentions this for perfect center via Albert in Thm 2.5 using the language of Brauer groups. $\endgroup$
    – Benjamin Steinberg
    Sep 11, 2014 at 19:58
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    $\begingroup$ The most basic noncommutative division rings are non-split quaternion algebras, which are of dimension 4 over their centers, and there are many non-split quaternion algebras in characteristic 2. A construction over the simplest nonperfect field of characteristic 2, $\mathbf F_2(t)$, is in exercise 11 of math.uconn.edu/~kconrad/ross2004/quatset4.pdf. $\endgroup$
    – KConrad
    Sep 12, 2014 at 2:31

1 Answer 1

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There are counterexamples for each $p$. The easiest maybe is the following: Let $F$ be the field of order $p^p$, and $\sigma$ be an automorphism of $F$ of order $p$. Let $D=F((t))$ be the set of Laurent series of the form $\sum a_it^i$ with the usual addition. Define multiplication by $ta=a^\sigma t$. Then $D$ is a division algebra with center $Z=\mathbb F_p((t^p))$, so $[F:Z]=p^2$.

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