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 $\\mathbb{R}$ 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).

单词	LocalField
释义	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 $\\mathbb{R}$ 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).
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