Let $K$ be a field, such as $\mathbb{K}(t_1,t_2,\cdots,t_n)$, where $t_1,\cdots t_n$ is the transcendence basis over $\mathbb{K}$.

How to characterize the Galois extension of it, including abelian case and nonabelian case, this is a similar question to Langlands program.

Partial results is that: when $K$ is finite field, or n=1, we know a lot about it. Using the Kato's method, we can get Artin map from Milnor $K_2$ group to fundamental group, however this is geometric Langlands program, which is established using the algebraic geometric tool. Galois group is replace by fundamental group, and class group is replace by Picard group or Chow group $\mathrm{CH}_0$ or Milnor K-group.

When $K$ is not finite field, can we get the similar theorem of class field theory for this field $\mathbb{K}(t_1,t_2,\cdots,t_n)$ as in classical class field theory without using the geometric tools?

I also notice geometric Langlands program for the loop group $G(\mathbb{C}((t)))$ (cf. Frankel), whether can we establish similar works for the group $G(\mathbb{C}((t_1,t_2,\cdots, t_n)))$. Can we get Langlands correspondence for the group $G(\mathbb{K}((t_1,t_2,\cdots, t_n)))$, where $K$ is arbitrary field?

Why the global function in the literature is often refer as the finite extension of $F_q(t)$, instead of the the finite extension of $\mathbb{K}(t_1,t_2,\cdots, t_n)$.

Thanks for any comments.

  • 3
    $\begingroup$ Asking to classify the Galois extensions of an arbitrary field seems way too broad to have any hope of admitting an answer, or at least one that could or would be given on MO. $\endgroup$
    – LSpice
    May 26 at 3:05
  • $\begingroup$ Welcome new contributor. You might sharpen your question by asking whether there is an analogue of the geometric Langlands correspondence for fields other than function fields of curves. $\endgroup$ May 26 at 10:45
  • $\begingroup$ @starr My focus is not geometric langlands $\endgroup$ May 26 at 14:13