Field generated by adjoining elements

Definition

Let $L/K$ be a field extension and let $S \subseteq L$. The field $K(S)$ is the intersection of all the subfields of $L$ that contain $K$ and $S$. It is the smallest intermediate field of $L/K$ that contains $S$.

Remarks

For $S = \{a_1, \dots, a_n\}$ one writes $K(a_1, \dots, a_n)$. The elements of $K(S)$ are the quotients $f(s_1, \dots, s_n)/g(s_1, \dots, s_n)$ with $f, g$ polynomials over $K$, $s_i \in S$ and $g(s_1, \dots, s_n) \neq 0$. One has $K(a, b) = K(a)(b)$. For example, $\mathbb{Q}(\sqrt{2}) = \{ p + q \sqrt{2} : p, q \in \mathbb{Q} \}$ inside $\mathbb{R}$.

Explanations

L1 human

To adjoin elements to a field, you take the smallest field that contains the old field and the new elements. It contains everything you can get from them by addition, subtraction, multiplication and division.