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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the number of field topologies on an infinite field


Author: John O. Kiltinen
Journal: Proc. Amer. Math. Soc. 40 (1973), 30-36
MSC: Primary 12J99
DOI: https://doi.org/10.1090/S0002-9939-1973-0318118-4
MathSciNet review: 0318118
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Abstract: K.-P. Podewski has recently proven that every countable infinite field admits $ {2^{{}_2{\aleph _0}}}$ different field topologies. Using methods of valuation theory, it is proven that every uncountable field, and more generally, every field $ F$ of infinite transcendence degree over some subfield, admits $ {2^{{2^{\vert F\vert}}}}$ field topologies. By purely set theoretic considerations, it then follows that there are $ {2^{{2^{\vert F\vert}}}}$ field topologies on any infinite field $ F$, no two of which are topologically isomorphic. This latter result is then generalized to any infinite commutative ring without proper zero-divisors. A further aspect of Podewski's work on countable fields is generalized in a final theorem which states that a field $ F$ of infinite transcendence degree admits $ {2^{{2^{\vert F\vert}}}}$ field topologies which fail to be suprema of locally bounded ring topologies.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0318118-4
Keywords: Number of field topologies, valuation, inductive ring topology, locally bounded, infinite transcendence degree, supremum of a family of topologies, finite intersection property with complements
Article copyright: © Copyright 1973 American Mathematical Society