= Solution
Let the monic <minimal polynomial> of $\alpha$ over $L$ have coefficients $a_0,\ldots,a_{r-1}$, and put $F=K(a_0,\ldots,a_{r-1})$. Each coefficient is algebraic over $K$, so adjoining these finitely many coefficients successively gives $[F:K]<\infty$. The same <polynomial> makes $\alpha$ algebraic over $F$, giving $[F(\alpha):F]\le r$. The <tower law for field extensions> yields $[F(\alpha):K]<\infty$. Its subfield $K(\alpha)$ therefore also has finite degree over $K$. This holds for every $\alpha\in M$, proving \b[$M/K$ is algebraic].
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