Regular split embedding problems over function fields of one variable over ample fields
The main result of the paper is the following: if \(K_0\) is an ample field, \(E_0\) is a function field of one variable over \(K_0\), and \(E\) is a finite Galois extension of \(E_0\), then each split (regular) \(K_0\)-embedding problem \[ G\rtimes G(E/E_0) \longrightarrow G(E/E_0) \] has a (regular) solution. This is a generalization of a theorem given by \textit{F. Pop} in an unpublished paper [The geometric case of a conjecture of Shafarevich, preprint, Heidelberg (1993)]. The paper is a continuation of a paper of the same two authors [Forum Math. 10, No. 3, 329-351 (1998; Zbl 0903.12003)], where they introduced a purely algebraic approach to embedding problems, using the ``patching principle''. The result stated above is obtained by a sequel of reduction steps: first, one can assume \(E_0=K_0(x)\), then \(K_0\) is replaced by its field of formal power series, so that it is complete under an ultrametric absolute value, and one can directly apply the patching principle.
scientific article; zbMATH DE number 1221659