Let $D$ be a division ring. I have in mind the following result.



> **Theorem**. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism of $E$. 


**Q1.** Is the Theorem correct? Any reference?


**Q2.** If $f$ fixes the centre of $D$ pointwise, can $E$ be chosen so as to have the same centre as $D$?



Extending an automorphism to an inner one
For Q1, you can form the skew polynomial ring $D[t;f]$ (i.e., the ring of polynomials in $t$ with coefficients from $D$, and multiplication satisfying $tr=f(r)t$ for $r\in D$). This is an Ore domain, and has a skew field of fractions $E$ consisting of formal fractions $rs^{-1}$ where $r,s\in D[t;f]$ and $s\neq0$. Then conjugation by $t$ induces the automorphism $f$ on $D$. 


This is fairly classical, and you can find details in (for example) Chapter 2 of P.M. Cohn's book "Skew Fields: Theory of General Division Rings".


I don't know the answer to Q2, but if no power of $f$ is inner, then I'd guess (but haven't checked) that the construction above works.


