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local field
A local field is a topological field which is Hausdorff and locally compact as a topological space.
Examples of local fields include:
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Any field together with the discrete topology.
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The field of real numbers.
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The field of complex numbers.
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The field of –adic rationals, or any finite extension thereof.
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The field of formal Laurent series in one variable with coefficients in the finite field of elements.
In fact, this list is complete—every local field is isomorphic as a topological field to one of the above fields.
1 Acknowledgements
This document is dedicated to those who made it all the way through Serre’s book [1] before realizing that nowhere within the book is there a definition of the term “local field.”
References
- 1 Jean–Pierre Serre, Local Fields, Springer–Verlag, 1979 (GTM 67).
Type of Math Object:
Definition
Major Section:
Reference
Groups audience:
Mathematics Subject Classification
13H99 None of the above, but in MSC2010 section 13Hxx12J99 None of the above, but in MSC2010 section 12Jxx
11S99 None of the above, but in MSC2010 section 11Sxx
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