Solutions to the matrix equation $\mathbf{AB-BA=I}$ over general fields Some days ago, I was thinking on a problem, which states that $$AB-BA=I$$ does not have a solution in $M_{n\times n}(\mathbb R)$ and $M_{n\times n}(\mathbb C)$. (Here $M_{n\times n}(\mathbb F)$ denotes the set of all $n\times n$ matrices with entries from the field $\mathbb F$ and $I$ is the identity matrix.)
Although I couldn't solve the problem, I came up with this problem:

Does there exist a field $\mathbb F$ for which that equation $AB-BA=I$ has a solution in $M_{n\times n}(\mathbb F)$?

I'd really appreciate your help.

HINT $\ $ Extending a 1936 result of Shoda for characteristic $0,$ Benjamin Muckenhoupt, a 2nd year graduate student of A. Adrian Albert, proved in the mid fifties that in the matrix algebra $\rm\ \mathbb M_n(F)\ $ over a field $\rm\:F\:$, a matrix $\rm\:M\:$  is a commutator $\rm\ M\: = \: A\:B - B\:A\ $ iff $\rm\:M\:$ is traceless, i.e. $\rm\ tr(M) = 0\:.$
From this we infer that $1$ is a commutator in  $\rm\mathbb M_n(\mathbb F_p)\ \iff\  n = tr(1) = 0\in \mathbb F_p \iff\ p\ |\ n  \:.$
Muckenhoupt and Albert's proof is short and simple and is freely acessible at the link below
A. A. Albert, B. Muckenhoupt. On matrices of trace zero,  Michigan Math. J. 4, #1 (1957), 1-3.
Tracing citations to this paper reveals much literature on representation by (sums of) commutators and tracefree matrices. For example, Rosset proved that in a matrix ring $\rm\:\mathbb M_n(R)\:$ over a commutative ring $\rm\:R\:,\:$ every matrix of trace zero is a sum of two commutators.
