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

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

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

3. Let $K$ be a field and denote by $\overline{K}$ 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	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