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If K is a field extension of F and $S\subseteq K$ is such that each s in S is F-algebraic, is it true that F[S] = F(S)?

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Yes. The only thing that really needs proving is that multiplicative inverses are there. If $s$ satisfies a polynomial, $a_0s^n+a_1s^{n-1}+\cdots+a_{n-1}s+a_n=0$, can you see how to get $s^{-1}$ as a polynomial in $s$?

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