lattice of fields
Let be a field and be its algebraic closure. The set of all intermediate fields (where ), ordered by set theoretic inclusion, is a poset. Furthermore, it is a complete lattice, where is the bottom and is the top.
This is the direct result of the fact that any topped intersection structure is a complete lattice, and is such a structure. However, it can be easily proved directly: for any collection of intermediate fields , the intersection is clearly an intermediate field, and is the infimum of the collection. The compositum of these fields, which is the smallest intermediate field such that , is the supremum of the collection.
It is not hard to see that is an algebraic lattice, since the union of any directed family of intermediate fields between and is an intermediate field. The compact elements in are the finite algebraic extensions of . The set of all compact elements in , denoted by , is a lattice ideal, for any subfield of a finite algebraic extension of is finite algebraic over . However, , as a sublattice, is usually not complete (take the compositum of all simple extensions , where are rational primes).
|Title||lattice of fields|
|Date of creation||2013-03-22 17:13:26|
|Last modified on||2013-03-22 17:13:26|
|Last modified by||CWoo (3771)|