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I'm reading the following theorem: enter image description here

My question is, why are $a_i$s in $A[\alpha_1...\alpha_n]$?

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    The as are the result of evaluating the elementary symmetric functions on the alphas. – 2017-02-27
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    I guess I know the answer now...how stupid I am... – 2017-02-27
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    I suggest you write a complete answer answering your question. – 2017-02-27

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There's an $\sigma$ sending $\alpha$ to $\alpha_i$ fixing $K$, hence $\alpha_i$ are integral over $A$ since $\alpha$ is, the rest is easy.