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**Embedding problems over large fields.**
*(English)*
Zbl 0862.12003

A field \(K\) is defined to be large if it has the property that every smooth curve over \(K\) has infinitely many \(K\)-rational points, provided that it has at least one rational point. Such fields include the PAC, PRC, and \(\text{P}_p \text{C}\) fields. This article is concerned with Galois theoretic properties of such fields. The first main theorem in this paper proves that every finite, split embedding problem for a large field \(K\) has proper, regular solutions. In particular, this means that every finite group \(G\) is realizable as a Galois group over the field \(K(t)\). This is then used to give a positive answer to a conjecture of Roquette, that the absolute Galois group of a countable, PAC, hilbertian field is profinite free. The proof of the first main theorem relies on the \({1\over 2}\) Riemann existence theorem with Galois action [the author, in Algebra and number theory, Proc. Conf. Essen 1992, 193-218 (1994; Zbl 0840.14012)] and uses the fact that a large field is existentially closed in the field of Laurent series over it.

The second main result is derived from the first and shows that every finite split embedding problem for a large hilbertian field has proper solutions. This is then used to give a positive answer to a semi-local version of Shafarevich’s conjecture, namely that for \({\mathcal P}\) a finite set of places of the global field \(K\), and \(K^{{\mathcal P}, cycl}\) the maximal cyclotomic extension of \(K^{\mathcal P}\), the absolute Galois group of \(K^{{\mathcal P}, cycl}\) is free. (The Shafarevich conjecture asserts that \(K^{cycl}\) is profinite free, and would imply the semilocal version.) Finally, the second main result is used to determine the Galois structure of the field \(K^{\mathcal P}\) of totally \({\mathcal P}\)-adic elements over a global field \(K\).

The second main result is derived from the first and shows that every finite split embedding problem for a large hilbertian field has proper solutions. This is then used to give a positive answer to a semi-local version of Shafarevich’s conjecture, namely that for \({\mathcal P}\) a finite set of places of the global field \(K\), and \(K^{{\mathcal P}, cycl}\) the maximal cyclotomic extension of \(K^{\mathcal P}\), the absolute Galois group of \(K^{{\mathcal P}, cycl}\) is free. (The Shafarevich conjecture asserts that \(K^{cycl}\) is profinite free, and would imply the semilocal version.) Finally, the second main result is used to determine the Galois structure of the field \(K^{\mathcal P}\) of totally \({\mathcal P}\)-adic elements over a global field \(K\).

Reviewer: Tara L.Smith (Cincinnati)