local field
A local field is a topological field which is Hausdorff and locally compact as a topological space
.
Examples of local fields include:
- •
Any field together with the discrete topology.
- •
The field of real numbers.
- •
The field of complex numbers
.
- •
The field of –adic rationals (http://planetmath.org/PAdicIntegers), or any finite extension
thereof.
- •
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).