Let K ⊆ L is a field extension and S ⊆ L be a non empty set. We define the subfield of L generated by K ∪ S, denoted by K(S), to be the smallest subfield of L containing K ∪ S. Moreover, K(S) is the intersection of all the subfields of L containing K ∪ S. In this case, we say that K(S) is the field obtained from K by adjoining the set S. Similarly, We define the subring of L generated by K ∪ S, denoted by K[S], to be the smallest subring of L containing K ∪ S. Clearly, K[S] is the intersection of all the subrings of L containing K ∪ S.
Why can we write this K ⊆ K[S] ⊆ K(S) ⊆ L.??????
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