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单词 LocalField
释义

local field


A local field is a topological field which is HausdorffPlanetmathPlanetmath and locally compact as a topological spaceMathworldPlanetmath.

Examples of local fields include:

  • Any field together with the discrete topology.

  • The field of real numbers.

  • The field of complex numbersMathworldPlanetmathPlanetmath.

  • The field p of p–adic rationals (http://planetmath.org/PAdicIntegers), or any finite extensionMathworldPlanetmath thereof.

  • The field 𝔽q((t)) of formal Laurent series in one variable t with coefficients in the finite fieldMathworldPlanetmath 𝔽q of q elements.

In fact, this list is completePlanetmathPlanetmath—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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更新时间:2025/5/4 6:55:53