algebraically dependent

Let $L$ be a field extension of a field $K$. Two elements $\alpha ,\beta$ of $L$ are algebraically dependent if there exists a non-zero polynomial $f(x,y)\in K[x,y]$ such that $f(\alpha ,\beta )=0$. If no such polynomial exists, $\alpha$ and $\beta$ are said to be algebraically independent.

More generally, elements ${\alpha }_1,\mathrm{\dots },{\alpha }_n\in L$ are said to be algebraically dependent if there exists a non-zero polynomial $f(x_1,\mathrm{\dots },x_n)\in K[x_1,\mathrm{\dots },x_n]$ such that $f({\alpha }_1,{\alpha }_2,\mathrm{\dots },{\alpha }_n)=0$. If no such polynomial exists, the collection of $\alpha$’s are said to be algebraically independent.

Title	algebraically dependent
Canonical name	AlgebraicallyDependent
Date of creation	2013-03-22 13:58:13
Last modified on	2013-03-22 13:58:13
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	8
Author	mathcam (2727)
Entry type	Definition
Classification	msc 12F05
Classification	msc 11J85
Related topic	DependenceRelation
Defines	algebraically independent
Defines	algebraic dependence
Defines	algebraic independence