It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar, Shreeram S. Abhyankar: Hilbert's Thirteenth Problem, Vitushkin, A. G. Vitushkin: On Hilbert's thirteenth problem and related questions, Chebotarev (N. G. Chebotarev, "On certain questions of the problem of resolvents") and others). It appears from one of Hilbert's papers [n 4] that this was his original intention for the problem.
The language of Hilbert there is "...Existenz von algebraischen Funktionen...",
i.e., "...existence of algebraic functions...".
As such, the problem is still unresolved.
It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar, Shreeram S. Abhyankar: Hilbert's Thirteenth Problem, Vitushkin, A. G. Vitushkin: On Hilbert's thirteenth problem and related questions, Chebotarev (N. G. Chebotarev, "On certain questions of the problem of resolvents") and others). It appears from one of Hilbert's papers [n 4] that this was his original intention for the problem.
The language of Hilbert there is "...Existenz von algebraischen Funktionen...",
i.e., "...existence of algebraic functions...".
As such, the problem is still unresolved.
