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I am studying transcendence bases and need a help on this problem.

Every algebraically independent subset of $F$ is contained in a transcendence basis.

Here are the relevant definitions:

Definition: Let $F$ be an extension field of $K$ and $S$ a subset of $F$. $S$ is algebraically dependent over $K$ if for some positive integer $n$ there exists a nonzero polynomial $f\in K[x_1, \cdots, x_n]$ such that $f(s_1,\cdots, s_n)=0$ for some distinct $s_1, \cdots, s_n\in S$. $S$ is algebraically independent over $K$ if $S$ is not algebraically dependent over $K.$

Definition: Let $F$ be an extension field of $K.$ A transcendence base of $F$ over $K$ is a subset $S$ of $F$ which is algebraically independent over $K$ and is maximal in the set of all algebraically independent subsets of $F$.

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    $\begingroup$ Do you know how to prove that every linearly independent set in a vector space can be extended to a basis for the vector space? Can you try to modify such a proof to fit the question you ask here? $\endgroup$ – Gerry Myerson Mar 26 at 2:00
  • $\begingroup$ @GerryMyerson Thanks very much for the idea! I solved the problem. $\endgroup$ – rgb12 Mar 26 at 2:26
  • $\begingroup$ Good! You can write it up, and post it as an answer. (But first maybe check the Related questions to see whether something like it has already been posted here.) $\endgroup$ – Gerry Myerson Mar 26 at 2:30
  • $\begingroup$ Are you still here, aloe? $\endgroup$ – Gerry Myerson Mar 27 at 11:35
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    $\begingroup$ I'm voting to close this question as off-topic because OP has lost interest. $\endgroup$ – Gerry Myerson Mar 28 at 11:49

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