algebraically dependentAlgebraicallyDependent2003-09-25 17:57:502006-02-24 19:57:30Definitionalgebraically independentalgebraic dependencealgebraic independence% this is the default PlanetMath preamble. as your knowledge
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\renewcommand{\>}{\rangle}Let $L$ be a field extension of a field $K$. Two elements $\alpha, \beta$ of $L$ are \emph{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 \emph{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.