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# 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.

Defines:

algebraically independent, algebraic dependence, algebraic independence

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DependenceRelation

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## Mathematics Subject Classification

12F05*no label found*11J85

*no label found*

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new question: Latent variable by adam_reith

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

new question: Lorenz system by David Bankom

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

## Comments

## elements from algebraic extension always algebraically depen...

Hi, maybe I'm confusing things but are't elements of an ALGEBRAIC field extension *always* algebraically dependent? Perhaps, L should rather be an *arbitrary* field extension of K...

Thanks