Algebraic Integers and Polynomial Roots

Question

Given that $k_1,...,k_n$ are algebraic integers and the complex number $c$ is a root of the polynomial $X^n+k_1X^{n-1}+..+k_n$ it is an exercise to prove that $c$ is also an algebraic integer. I know that, by definition, an algebraic integer is a complex number that is a root of some monic polynomial with integer coefficients. My question then is I don't quite get what is there exactly to be proved here. Am I missing something?

Edit: So, can I say for any polynomial $f$ with algebraic coefficients and root $c$, there is a polynomial $g$ with integer coefficients and with same root $c$. Then perhaps multiplying some polynomials with some roots will give a polynomial with integer coefficients? But I am not sure of the details~

Knowing that something is a root of a polynomial with algebraic integer coefficients does not instantly prove that it is a root of a polynomial with integer coefficients. — lulu, Mar 26 '17 at 15:51
The definition of algebraic integer requires that the $k_i$ be (rational) integers. So there,is something to prove. — hardmath, Mar 26 '17 at 15:52
Thanks for the clarification, I have edited my post according to my new understanding~ — Homaniac, Mar 26 '17 at 16:03
In general you need to do more than multiply by a constant. After all you want to get a monic polynomial. The result you are after here is usually approached by field extensions rather than by direct polynomial construction. — hardmath, Mar 26 '17 at 16:17
I am not an expert on this topic, but this looks related to me: Can every polynomial with algebraic coeffitients be transformed into a polymonial with integer coefficients and at least some of the same roots? — Martin R, Mar 26 '17 at 16:20
Indeed and that is all rather abstract hm..could you outline this 'field extension' approach in more details? — Homaniac, Mar 26 '17 at 16:20

Algebraic Integers and Polynomial Roots

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