algebraic extension
Definition 1.
Let $L\mathrm{/}K$ be an extension of fields. $L\mathrm{/}K$ is said to be an algebraic extension^{} of fields if every element of $L$ is algebraic 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 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
$${X}^{2}2qX+{q}^{2}2{t}^{2}=0$$ 
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.
Let $K$ be a field and denote by $\overline{K}$ the algebraic closure^{} of $K$. Then the extension $\overline{K}/K$ is algebraic.

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.
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  20130322 13:57:27 
Last modified on  20130322 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 