Composite field (mathematics)
A composite field or compositum of fields is an object of study in field theory. Let K be a field, and let, be subfields of K. Then the composite of and is the field defined as the intersection of all subfields of K containing both and. The composite is commonly denoted.
Properties
Equivalently to intersections we can define the composite to be the smallest subfield of K that contains both and. While for the definition via intersection, well-definedness hinges only on the property that intersections of fields are themselves fields, here two auxiliary assertions are included. Firstly, that there exist [Maximal and Maximal and minimal elements|minimal elements|minimal] subfields of K that include and and secondly, that such a minimal subfield is unique and therefore justly called the smallest.It also can be defined using field of fractions
where is the set of all -rational expressions in finitely many elements of.
Let be a common subfield and a Galois extension then and are both also Galois and there is an isomorphism given by restriction
For finite field extension this can be explicitly found in Milne and for infinite extensions this follows since infinite Galois extensions are precisely those extensions that are unions of an set of finite Galois extensions.
If additionally is a Galois extension then and are both also Galois and the map
is a group homomorphism which is an isomorphism onto the subgroup
See Milne.
Both properties are particularly useful for and their statements simplify accordingly in this special case. In particular is always an isomorphism in this case.