请输入您要查询的字词:

 

单词 LatticeOfFields
释义

lattice of fields


Let K be a field and K¯ be its algebraic closureMathworldPlanetmath. The set Latt(K) of all intermediate fields E (where KEK¯), ordered by set theoretic inclusion, is a poset. Furthermore, it is a complete latticeMathworldPlanetmath, where K is the bottom and K¯ is the top.

This is the direct result of the fact that any topped intersection structure is a complete lattice, and Latt(K) is such a structureMathworldPlanetmath. However, it can be easily proved directly: for any collectionMathworldPlanetmath of intermediate fields {EiiI}, the intersectionMathworldPlanetmath is clearly an intermediate field, and is the infimumMathworldPlanetmathPlanetmath of the collection. The compositum of these fields, which is the smallest intermediate field E such that EiE, is the supremumMathworldPlanetmath of the collection.

It is not hard to see that Latt(K) is an algebraic lattice, since the union of any directed family of intermediate fields between K and K¯ is an intermediate field. The compact elements in Latt(K) are the finite algebraic extensionsMathworldPlanetmath of K. The set of all compact elements in Latt(K), denoted by LattF(K), is a lattice ideal, for any subfieldMathworldPlanetmath of a finite algebraic extension of K is finite algebraic over K. However, LattF(K), as a sublattice, is usually not completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (take the compositum of all simple extensions (p), where p are rational primes).

随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 12:07:23