A composite field or compositum of fields is an object of study in field theory. Let K be a field, and let E_1, E_2 be subfields of K. Then the (internal) composite of E_1 and E_2 is the field defined as the intersection of all subfields of K containing both E_1 and E_2. The composite is commonly denoted E_1E_2.

Properties

Equivalently to intersections we can define the composite E_1E_2 to be the smallest subfield of K that contains both E_1 and E_2. While for the definition via intersection well-definedness hinges only on the property that intersections of fields are themselves fields, here two auxiliary assertion are included. That 1. there exist minimal subfields of K that include E_1 and E_2 and 2. that such a minimal subfield is unique and therefor justly called the smallest. It also can be defined using field of fractions where F(S) is the set of all F-rational expressions in finitely many elements of S. Let be a common subfield and E_1/L a Galois extension then E_1E_2/E_2 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 (infinite) set of finite Galois extensions. If additionally E_2/L is a Galois extension then E_1E_2/L 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 \psi is always an isomorphism in this case.

External composite

When E_1 and E_2 are not regarded as subfields of a common field then the (external) composite is defined using the tensor product of fields. Note that some care has to be taken for the choice of the common subfield over which this tensor product is performed, otherwise the tensor product might come out to be only an algebra which is not a field.

Generalizations

If is a set of subfields of a fixed field K indexed by the set I, the generalized composite field can be defined via the intersection