Let be a tower of fields. If and are algebraic, then is algebraic.

**Proof:**

(Hungerford pg 237, reworded)

Let . Since is algebraic over , there exists some () such that

Let , then is algebraic over . Hence is finite. Note that is finitely generated and algebraic, since each is algebraic over . Thus is finite.

Thus by Tower Law, is finite, thus algebraic.

Hence is algebraic over . Since was arbitrary, is algebraic over .

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