2
$\begingroup$

For a fixed positive integer $n$, the group $G=GL_n(\mathbb{C})$ acts on the field $K=\mathbb{C}(t_1,\ldots,t_n)$ by linear change of variables. I would like to know if there is something like a Galois correspondence for this action. Specifically:

Is there a description of which subfields of $K$ are of the form $K^H$ for some subgroup $H\leq G$?

Is there a description of which subgroups of $G$ are of the form $$ G_L:=\{g\in G:g|_L=\mathrm{Id}_L\} $$ for some subfield $L\subset K$?

A few observations: For the first question, any subfield of the form $K^H$ contains $\mathbb{C}$ and is invariant under the diagonal action of $\mathbb{C}^\times$. I wonder if these conditions are sufficient.

For the second question, every subgroup of the form $G_L$ is Zariski closed, but this is not sufficient. For instance, the subgroup of diagonal matrices cannot be the stabilizer of any subfield, since anything fixed by the diagonal is fixed by all of $G$.

$\endgroup$
1
  • 1
    $\begingroup$ One subgroup of your $G$ is $S_n$, acting on the variables $t_i$ as permutations. The nature of the fixed field of your $K$ by subgroups of $S_n$ is subtle. See a theorem of Swan at en.wikipedia.org/wiki/Rational_variety (note $K$ there is your $\mathbf C$, and $L$ there is your $K$, more or less). $\endgroup$
    – KConrad
    Jan 15, 2014 at 19:54

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.