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Abstract: A central conjecture in modern Galois theory asserts that every finite group occurs as a Galois group over the field of rational functions K(x), for every field K. While in general the conjecture is wide open, for large classes of important fields it is known to hold, most notably whenever K is complete with respect to an absolute value. The "archimedean" case of this theorem (where K is the field of real or complex numbers) is a consequence of Riemann's Existence Theorem and methods of complex topology. The non-archimedean case (where K is a local field) was proven in 1984 by Harbater, using methods of formal geometry, most notably Grothendieck's Existence Theorem.
Harbater's constructions draw inspriation from the mentioned complex topological methods, but the analogy between these situations is quite limited and not fully understood. In this work we present a new approach which allows for a uniform proof of the theorem (and more). The central idea is to use a new type of objects in order to patch Galois groups -- Wiener Algebras (an object arising from Harmonic analysis). The proof via this approach is essentially elementary, relying on no deep theorems.
No background is assumed.