Let $K / F$ be a field extension and $S$ be a subset of $K$. Let $$ F(S) = \text{the intersection of all subfields of } K \text{ which contains }F\text{ and }S. $$ One can check that $F(S)$ is a subfield of $K$ and is the smallest subfield of $K$ that contains both $F$ and $S$. We call $F(S)$ the field generated by $S$ over $F$.

Given $S \subseteq K$, taking intersections is a general way to generate a smaller field extension of $F$, which also gives the smallest one that contains both $S$ and $F$.