Geodesic
A completion of $\mathbb{Z}$ is a field
Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 689-706 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence $\mathbb{L}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/(p)$. Further, we show that $(\mathbb{Z}, \mathbb{L}^\ast _1)$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.
We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence $\mathbb{L}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/(p)$. Further, we show that $(\mathbb{Z}, \mathbb{L}^\ast _1)$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.
Classification : 13J10, 13J99, 54A20, 54H13
Keywords: sequential convergence; convergence ring; completion of a convergence ring 
@article{CMJ_2003_53_3_a15,
     author = {Marcos, J. E.},
     title = {A completion of $\mathbb{Z}$ is a field},
     journal = {Czechoslovak Mathematical Journal},
     pages = {689--706},
     year = {2003},
     volume = {53},
     number = {3},
     mrnumber = {2000063},
     zbl = {1080.54500},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a15/}
}
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Marcos, J. E. A completion of $\mathbb{Z}$ is a field. Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 689-706. http://geodesic.mathdoc.fr/item/CMJ_2003_53_3_a15/