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Let $F \subset K$ be a field extension of degree $n$, then $F^n \cong K$ as $F$-vectorspaces. Now $K^\times$ acts on $F^n$, by multiplication on $K$, and so $K^\times$ embeds into $GL_n(F)$, and every Galois element gives an automorphism of $K^\times$.

Question: Under which conditions can it be extended to an automorphism of $GL_n(F)$? How?

Cyclic extension, abelian extension, solvalabe extension, general extension?

I am mostly interested in the case, where $F$ is a local field.

Example $\mathbb{R} \subset \mathbb{C}$: We fix an $\mathbb{R}$-basis $\\{ 1,i \\}$. The multipliaction by $a+ib$ correspond to the matrix $ \begin{pmatrix} a & -b \newline b & a \end{pmatrix}.$ The Galois element is complex conjugation and corresponds to transpose on the above matrices. Can it be extended to the group $GL_2(\mathbb{C})$?

Motivation: I am actually hoping for an explanation of the Caley transform introduced here on page 2: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=162025&vfpref=html&r=28&mx-pid=237707

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    You are welcome! As a meta remark: if you delete comments that others have responded to, then you leave other peoples' comments orphaned, which is not very nice. E.g. now, the second sentence of my previous comment seems to come out of nowhere.2012-04-25

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Maybe I am wrong, but consider rather the more canonical group $\mathrm{Aut}_F(K)$ of $F$-linear automorphisms on $K$. The Galois group naturally embedds and acts as inner automorphisms. The elements of $K^\times$ correspond to homothecies and the (inner) Galos action on the homothecies correspond to the natural action on $K$.