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  E 1 {\displaystyle E_{1}} ,  E 2 {\displaystyle E_{2}} be subfields of K. Then the (internal) composite[1] of  E 1 {\displaystyle E_{1}} and  E 2 {\displaystyle E_{2}} is the field defined as the intersection of all subfields of K containing both  E 1 {\displaystyle E_{1}} and  E 2 {\displaystyle E_{2}} . The composite is commonly denoted  E 1 E 2 {\displaystyle E_{1}E_{2}} .

## Properties

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Equivalently to intersections we can define the composite  E 1 E 2 {\displaystyle E_{1}E_{2}} to be the smallest subfield[2] of K that contains both  E 1 {\displaystyle E_{1}} and  E 2 {\displaystyle 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 assertions are included. Firstly, that there exist minimal subfields of K that include  E 1 {\displaystyle E_{1}} and  E 2 {\displaystyle E_{2}} and secondly, that such a minimal subfield is unique and therefore justly called the smallest.

It also can be defined using field of fractions

E 1 E 2 = E 1 ( E 2 ) = E 2 ( E 1 ) , {\displaystyle E_{1}E_{2}=E_{1}(E_{2})=E_{2}(E_{1}),}

where  F ( S ) {\displaystyle F(S)} is the set of all  F {\displaystyle F} -rational expressions in finitely many elements of  S {\displaystyle S} .[3]

Let  L ⊆ E 1 ∩ E 2 {\displaystyle L\subseteq E_{1}\cap E_{2}} be a common subfield and  E 1 / L {\displaystyle E_{1}/L} a Galois extension then  E 1 E 2 / E 2 {\displaystyle E_{1}E_{2}/E_{2}} and  E 1 / ( E 1 ∩ E 2 ) {\displaystyle E_{1}/(E_{1}\cap E_{2})} are both also Galois and there is an isomorphism given by restriction

Gal ( E 1 E 2 / E 2 ) -> Gal ( E 1 / ( E 1 ∩ E 2 ) ) , σ ↦ σ | E 1 . {\displaystyle {\text{Gal}}(E_{1}E_{2}/E_{2})\rightarrow {\text{Gal}}(E_{1}/(E_{1}\cap E_{2})),\sigma \mapsto \sigma |_{E_{1}}.}

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  E 2 / L {\displaystyle E_{2}/L} is a Galois extension then  E 1 E 2 / L {\displaystyle E_{1}E_{2}/L} and  ( E 1 ∩ E 2 ) / L {\displaystyle (E_{1}\cap E_{2})/L} are both also Galois and the map

ψ : Gal ( E 1 E 2 / L ) -> Gal ( E 1 / L ) × Gal ( E 2 / L ) , σ ↦ ( σ | E 1 , σ | E 2 ) {\displaystyle \psi :{\text{Gal}}(E_{1}E_{2}/L)\rightarrow {\text{Gal}}(E_{1}/L)\times {\text{Gal}}(E_{2}/L),\sigma \mapsto (\sigma |_{E_{1}},\sigma |_{E_{2}})}

is a group homomorphism which is an isomorphism onto the subgroup

H = { ( σ 1 , σ 2 ) : σ 1 | E 1 ∩ E 2 = σ 2 | E 1 ∩ E 2 } = Gal ( E 1 / L ) × Gal ( ( E 1 ∩ E 2 ) / L ) Gal ( E 2 / L ) ⊆ Gal ( E 1 / L ) × Gal ( E 2 / L ) . {\displaystyle H=\\{(\sigma _{1},\sigma _{2}):\sigma _{1}|_{E_{1}\cap E_{2}}=\sigma _{2}|_{E_{1}\cap E_{2}}\\}={\text{Gal}}(E_{1}/L)\times _{{\text{Gal}}((E_{1}\cap E_{2})/L)}{\text{Gal}}(E_{2}/L)\subseteq {\text{Gal}}(E_{1}/L)\times {\text{Gal}}(E_{2}/L).}

See Milne.[6]

Both properties are particularly useful for  L = E 1 ∩ E 2 {\displaystyle L=E_{1}\cap E_{2}} and their statements simplify accordingly in this special case. In particular  ψ {\displaystyle \psi } is always an isomorphism in this case.

## External composite

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When  E 1 {\displaystyle E_{1}} and  E 2 {\displaystyle E_{2}} 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.

## Generalizations

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If  E = { E i : i ∈ I } {\displaystyle {\mathcal {E}}=\left\\{E_{i}:i\in I\right\\}} 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

⋁ i ∈ I E i = ⋂ F ⊆ K s.t. ∀ i ∈ I : E i ⊆ F F . {\displaystyle \bigvee _{i\in I}E_{i}=\bigcap _{F\subseteq K{\text{ s.t. }}\forall i\in I:E_{i}\subseteq F}F.}

## Notes

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1. ^ Roman, p. 42.
2. ^ Roman, p. 42.
3. ^ Lubin, Jonathan. "The elements in the composite field FK".
4. ^ Milne, p. 40; take into account the preliminary definition of Galois as finite on p. 37
5. ^ Milne, p. 93 and 99
6. ^ Milne, p. 41 and 93
7. ^ "Compositum", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
8. ^ Roman, p. 42.

## References

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* Roman, Steven (2006). Field Theory. GTM. Vol. 158\. New York: Springer-Verlag. doi:10.1007/0-387-27678-5. ISBN 978-0-387-27677-9., especially chapter 2
* "Compositum", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
* Milne, James S. (2022). Fields and Galois Theory (v5.10).

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