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 $F$.