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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 $D$?

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    $\begingroup$ Why doesn't Skolem-Noether work? $\endgroup$ Sep 7, 2017 at 19:11
  • $\begingroup$ @WatsonLadd Don't you need finite dimensionality over the center for Skolem-Noether? To the OP: any conditions on D? $\endgroup$
    – Kimball
    Sep 7, 2017 at 22:57
  • $\begingroup$ If I am not mistaken, Skolem-Noether would apply if D was finite dimensional over its centre (in this case, take $E=D$ as $f$ is inner). What if it is not? $\endgroup$
    – Drike
    Sep 7, 2017 at 22:59
  • $\begingroup$ @Kimball I'm afraid not, no condition on $D$. $\endgroup$
    – Drike
    Sep 7, 2017 at 23:01

1 Answer 1

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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.

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  • $\begingroup$ Thanks for the answer. Why not consider directly the division ring of Laurent series $D((t,f))$ ? $\endgroup$
    – Drike
    Sep 8, 2017 at 11:05
  • $\begingroup$ @Drike Yes, that's easier. $\endgroup$ Sep 8, 2017 at 11:14
  • $\begingroup$ You are right about your last comment: if no power of $f$ is inner and $f$ fixes $Z(D)$ pointwise, then $D((t,f))$ has centre $Z(D)$. $\endgroup$
    – Drike
    Sep 17, 2017 at 6:20

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