PlanetMath: algebraic extension

(more info)

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algebraic extension

(Definition)

Definition 1 Let be an extension of fields. is said to be an algebraic extension of fields if every element of is algebraic over . If is not algebraic then we say that it is a transcendental extension of fields.

Examples:

Let $L=\mathbb{Q}(\sqrt{2})$ . The extension $L/\mathbb{Q}$ is an algebraic extension. Indeed, any element $\alpha\in L$ is of the form
$\displaystyle \alpha=q+t\sqrt{2}\in L$
for some $q,t\in\mathbb{Q}$ . Then $\alpha\in L$ is a root of
$\displaystyle X^2-2qX+q^2-2t^2=0$
The field extension $\mathbb{R}/ \mathbb{Q}$ is not an algebraic extension. For example, $\pi\in \mathbb{R}$ is a transcendental number over $\mathbb{Q}$ (see pi). So $\mathbb{R}/\mathbb{Q}$ is a transcendental extension of fields.
Let be a field and denote by $\overline{K}$ the algebraic closure of . Then the extension $\overline{K}/K$ is algebraic.
In general, a finite extension of fields is an algebraic extension. However, the converse is not true. The extension $\overline{\mathbb{Q}}/\mathbb{Q}$ is far from finite.
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 $p(x)\in \mathbb{Q}[x]$ .