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Let $F/K$ be an extension field and let $D$ be a subset of $F$ and $z \in K(D)$. Why we can find a subset $\{d_{1},d_{2},...,d_{n}\} \subseteq D$ such that $z \in K(d_{1},d_{2},...,d_{n})$?

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Unpack what it means for $z$ to be in $K(D)$. –  Qiaochu Yuan Apr 7 '11 at 3:58
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In general, operations in algebra must be finite - we cannot take arbitrary sums or products of elements (in analysis, the concept of limit is available, in which case we define infinite sums and products via limits of finite ones). Thus, consider that $K(D)$ is the set of elements of $F$ that can be gotten by some algebraic combination (finite sums and products) of elements of $K$ and $D$. The specific element $z\in K(D)$ can only use finitely many elements of $D$.

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