Let be a field and let be algebraic over . Prove that is the composite of the fields .
Recall that the set of all algebraic elements over is a field, and we consider finite extensions to be subfields of . Then the composite of the fields is the inclusion-smallest subfield of containing all the . (The intersection of subfields is a subfield, so this is just the intersection over the class of subfields containing the .)
Certainly is a subfield containing each of the . Now if is a field containing the , then contains by definition. ( is the smallest subfield containing and , for any set .)
So is indeed the composite of the fields .