Smallest subfield containing F and {a}

Question

Let F be a field a E be an it's extension field, let a belongs to E then we define F(a) to be the smallest subfield of E containg both F and {a}. My question here is that, from collection of all the subfields of E containg both F and {a}, How can we choose a smallest one?[because( according to me), we can only compare two sets if either of them is contained in other]

Answer 1

You can indeed only compare two fields if one of them is contained in the other. In fact, there is a subfield of $E$ that is the smallest subfield of $E$ containing $F$ and $a$ in the sense that it is indeed contained in every other subfield of $E$ containing $F$ and $a$.

You can actually construct it by taking the intersection of all subfields of $E$ containing $F$ and $a$. This obviously is contained in all those subfields and it obviously contains $F$ and $a$; the point to verify is that it is a field.