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 $ℝ$ of real numbers.

• The field $ℂ$ of complex numbers.

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

• The field ${𝔽}_q((t))$ of formal Laurent series in one variable $t$ with coefficients in the finite field ${𝔽}_q$ 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.”

Title	local field
Canonical name	LocalField
Date of creation	2013-03-22 12:48:07
Last modified on	2013-03-22 12:48:07
Owner	djao (24)
Last modified by	djao (24)
Numerical id	7
Author	djao (24)
Entry type	Definition
Classification	msc 13H99
Classification	msc 12J99
Classification	msc 11S99