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Inspired by a recent paper of Alain Connes and Caterina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces (“Singer actions”), we consider several fundamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields F the plane P 2 ( F ) = PG ( 2 , F ) (and more generally the space P n ( F ) = PG ( n , F ) ) admits a Singer group, and show, e.g., that for any prime p and any positive integer n > 1 , PG ( n , F p ‾ ) cannot admit Singer groups ( F p ‾ an algebraic closure of F p ). One of the main results in characteristic 0, which is a corollary of a criterion which applies to many other fields, is that PG ( m , R ) with m ≠ 0 a positive even integer, cannot admit Singer groups. [ABSTRACT FROM AUTHOR]

Abstract: According to Markov (1946) , a subset of an abelian group G of the form , for some integer n and some element , is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian topology on G called the Zariski, or verbal, topology of G; see Bryant (1977) . We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet–Urysohn. For a countable family of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology on G such that the -closure of each member of coincides with its -closure. As an application, we provide a characterization of the subsets of G that are -dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov (1946) . [ABSTRACT FROM AUTHOR]

Abstract: A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]

We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil–Rödl Theorem, the Ramsey property of partial orders and metric spaces as well as the authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered S -metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the structural Ramsey theorem. Both of these results are natural, and easy to state, yet their proofs involve most of the theory developed here. We also characterise Ramsey lifts of classes of structures defined by finitely many forbidden homomorphisms and extend this to special cases of classes with closures. This has numerous applications. For example, we find Ramsey lifts of many Cherlin–Shelah–Shi classes. [ABSTRACT FROM AUTHOR]