In my research, I've come twice now towards a skew field $K$ that satisfies the following:



> $$\text{for all non-central element $a$, the map }\quad x\mapsto ax-xa\quad\text{ is onto.}$$


I am hoping such a field does not exist but am unable to prove it.


**Does this condition ring a bell** to some of you ?


**Any example of such a field** in both zero and non-zero characteristic ?


**Edit**: reading the paper suggested by @TomDeMedts'comment made me realise that this is precisely Question 3 of <http://www.ams.org/journals/proc/1961-012-03/S0002-9939-1961-0124355-6/> asked by G. Meisters in 1961.



Existence of a skew field with surjective inner derivations
In *The range of derivations on a skew field and the equation ax-xb=c*, Journal of the Indian Math. Soc. 37 (1973), 61--69, P.M. Cohn shows the following :



> **Theorem (Cohn 1973).** Given a field $K$ with centre $k$, there is an extension field $L$ (still with $k$ as centre) such that the equation $$ax-xa=c$$ has a solution for each $a\in L$ that is transcendental over $k$ and each $c\in L$.


In particular, taking a skew field $K$ having an algebraically closed centre (hence infinite dimentionnal over $k$ as @René notes) provides an $L$ for which every non-zero inner derivation is onto.


