# local field

Examples of local fields include:

• Any field together with the discrete topology.

• The field $\mathbb{R}$ of real numbers.

• The field $\mathbb{C}$ of complex numbers.

• The field $\mathbb{Q}_{p}$ of $p$–adic rationals (http://planetmath.org/PAdicIntegers), or any finite extension thereof.

• The field $\mathbb{F}_{q}((t))$ of formal Laurent series in one variable $t$ with coefficients in the finite field $\mathbb{F}_{q}$ of $q$ 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).
Title local field LocalField 2013-03-22 12:48:07 2013-03-22 12:48:07 djao (24) djao (24) 7 djao (24) Definition msc 13H99 msc 12J99 msc 11S99