A composite field or compositum of fields is an object of study in field theory.
be subfields of K. Then the (internal) composite[1] of
is the field defined as the intersection of all subfields of K containing both
The composite is commonly denoted
Equivalently to intersections we can define the composite
to be the smallest subfield[2] of K that contains both
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
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
-rational expressions in finitely many elements of
be a common subfield and
a Galois extension then
are both also Galois and there is an isomorphism given by restriction For finite field extension this can be explicitly found in Milne[4] 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.
[5] If additionally
is a Galois extension then
are both also Galois and the map is a group homomorphism which is an isomorphism onto the subgroup See Milne.
[6] Both properties are particularly useful for
and their statements simplify accordingly in this special case.
is always an isomorphism in this case.
are not regarded as subfields of a common field then the (external) composite is defined using the tensor product of fields.
[7] 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.
is a set of subfields of a fixed field K indexed by the set I, the generalized composite field[8] can be defined via the intersection
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