Simple field extension

Definition. A field extension $L/K$ is simple if there exists $\alpha\in L$ such that

the smallest subfield of $L$ containing $K$ and $\alpha$. One also says “$L$ is obtained from $K$ by adjoining $\alpha$”.

If $\alpha$ is algebraic over $K$, then $K(\alpha)\cong K[x]/(m_\alpha(x))$ where $m_\alpha$ is the minimal polynomial of $\alpha$.

See also. finitely generated field extension , transcendental element .

Examples.

1. $\mathbb{Q}(\sqrt2)/\mathbb{Q}$ is simple with generator $\sqrt2$.
2. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is simple, where $\zeta_n$ is a primitive n-th root of unity .
3. If $t$ is an indeterminate, then $\mathbb{Q}(t)/\mathbb{Q}$ is simple and $t$ is transcendental over $\mathbb{Q}$.