# 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 $\alpha_{1},\ldots,\alpha_{n}\in L$ are said to be algebraically dependent if there exists a non-zero polynomial $f(x_{1},\ldots,x_{n})\in K[x_{1},\ldots,x_{n}]$ such that $f(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})=0$. If no such polynomial exists, the collection of $\alpha$’s are said to be algebraically independent.

Title algebraically dependent AlgebraicallyDependent 2013-03-22 13:58:13 2013-03-22 13:58:13 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 12F05 msc 11J85 DependenceRelation algebraically independent algebraic dependence algebraic independence