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An algebraic integer is a complex number that is the root of monic polynomial with integer coefficients. Show that the set of algebraic integers is a subring of $C$. (Hint: Use symmetric function theorem).

I also know that $\alpha \in$ $C$ is an algebraic integer if and only if $m_\alpha,_Q \in Z[x].$

Thanks for the help.

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    The most explicit proof that $\overline{\mathbb{Z}}$ is a ring is with the [resultant](https://en.wikipedia.org/wiki/Resultant#Number_theory). I'm not sure what means using the symmetric function theorem, can you elaborate on what you proved earlier ? – 2017-02-07
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    what is the meaning of $m_\alpha,_Q $? – 2017-02-07
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    $m_\alpha,_Q$ is minimal polynomial of $\alpha$ over field $Q$ – 2017-02-07
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    This link may help you, http://math.stackexchange.com/questions/155122/how-to-prove-that-the-sum-and-product-of-two-algebraic-numbers-is-algebraic – 2017-02-07

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