Let F be i field with subfields K,L. Prove that there is a largest subfield of F contained in both K and L, and a smallest subfield of containung both K and L.

Rui Baldwin 2020-12-15 Answered
Let F be i field with subfields K,L. Prove that there is a largest subfield of F contained in both K and L, and a smallest subfield of containung both K and L.
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Expert Answer

Mitchel Aguirre
Answered 2020-12-16 Author has 94 answers

To get the general picture, first consider a simple example.
F,K,L fields, with K,L sub F (subfields)
Consider the followinf wxample: F=R,K=Q(2),L=Q(3). Then,
1) the largest subfield of F contained in both KandLisQ=KL
2) the smallest subfield of F=R containing both KandLisQ(2,3)
Now consider the general case. The largest (sub)field of F contained in both K and L is clearly KL. See details in the proof.
Let F,K,L be fields, with K,LF (subfields)
Let M=KL,
Claim: M is the largest sunfield contained in both K and L.
Proof: To start wi M is a field.
So, a subfield of F,SK,SLSKL=(M)
This, shows that M=K cap L is the maximal subfield of F with respect to this property. Hence the claim is true.
To show the existence of the smallest subfield containing K and L , proceed as in the proof.. Note that we cant work with the union of K and L , as the union of two fields need not be a field. So, a different approach is needed.
Let F,K,L be fields with K,LF (subfields)
Claim: m, m the smallest subfield of F, containing both K and L (as subfields)
Proofs: Define family S={P:P a subfield of F, P contains both K and L}
To start with, note that S is non - empty family, as F in S
Completing the proof that M , as defined, is the smallest subfield of F containing both K and L.
Proof: Define the family S={P:P a subfield of F, P contains both K and L}
To start with, note that S is non - empty family, asFS
Define M=PSP.
As P contains both KandLPS, M contains both K and L.
M is the intersection of all such P, M is the minimal among all subfields of F containg both K and L
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