algebraic extension

Definition 1.

Let $L\mathrm{/}K$ be an extension of fields. $L\mathrm{/}K$ is said to be analgebraic extension of fields if every element of $L$ isalgebraic over $K$. If $L\mathrm{/}K$ is not algebraic then we say that it is a transcendental extension of fields.

Examples:

1. Let $L=\mathbb{Q}(\sqrt{2})$. The extension $L/\mathbb{Q}$ is analgebraic extension. Indeed, any element $\alpha \in L$ is of theform

   $$\alpha =q+t\sqrt{2}\in L$$

   for some $q,t\in \mathbb{Q}$. Then $\alpha \in L$ is a root of

   $$X^2-2qX+q^2-2t^2=0$$

2. The field extension $ℝ/\mathbb{Q}$ is not an algebraicextension. For example, $\pi \in ℝ$ is a transcendental numberover $\mathbb{Q}$ (see pi). So $ℝ/\mathbb{Q}$ is a transcendental extension of fields.

3. Let $K$ be a field and denote by $\overline{K}$ thealgebraic closure of $K$. Then the extension $\overline{K}/K$ isalgebraic.

4. In general, a finite extension of fields is an algebraicextension. However, the converse is not true. The extension$\overline{\mathbb{Q}}/\mathbb{Q}$ is far from finite.

5. The extension $\mathbb{Q}(\pi )/\mathbb{Q}$ is transcendental because $\pi$ is a transcendental number, i.e. $\pi$ is not the root of any polynomial $p(x)\in \mathbb{Q}[x]$.