# algebraic extension

###### Definition 1.

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

Examples:

1. 1.

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

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

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

 $X^{2}-2qX+q^{2}-2t^{2}=0$
2. 2.

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.

3. 3.

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

4. 4.

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.

5. 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  $p(x)\in\mathbb{Q}[x]$.

 Title algebraic extension Canonical name AlgebraicExtension Date of creation 2013-03-22 13:57:27 Last modified on 2013-03-22 13:57:27 Owner alozano (2414) Last modified by alozano (2414) Numerical id 7 Author alozano (2414) Entry type Definition Classification msc 12F05 Synonym algebraic field extension Related topic Algebraic Related topic FiniteExtension Related topic AFiniteExtensionOfFieldsIsAnAlgebraicExtension Related topic ProofOfTranscendentalRootTheorem Related topic EquivalentConditionsForNormalityOfAFieldExtension Defines examples of field extension Defines transcendental extension