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).

Title lattice of fields
Canonical name LatticeOfFields
Date of creation 2013-03-22 17:13:26
Last modified on 2013-03-22 17:13:26
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 12F99