Coding complete theories in Galois groups. (English) Zbl 1146.03019

For a topological space \(X\) we denote by \(\mathcal{C}(X)\) the set of closed, non-empty subsets of \(X.\) The Vietoris space of \(X,\) denoted \(\mathcal{V}(X)\), is the topology on \(\mathcal{C}(X)\) with basic open sets \(\langle U_1,\dots,U_{n}\rangle:= \{C\in \mathcal{C}(X):C\subseteq \bigcup U_{i},i=1,\dots,k\) and \( C\cap U_{i}\neq \emptyset \) for \(i=1,\dots,k\}\) for each finite collection \( U_{1},\dots,U_{n}\) of open sets of \(X.\) In this paper the author give a new characterization of the spaces of complete theories of pseudo-finite fields and of algebraically closed fields with a generic automorphism in terms of the Vietoris topology on absolute Galois groups of prime fields.


03C60 Model-theoretic algebra
12L12 Model theory of fields
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