This is a tut question, where beforehand I had to prove that if $p$ is prime and $a$ is an integer and if $p|a^n$ then $p|a$.
And from this I am supposed to prove that if $\alpha$ is a zero of the monic polynomial $T(x) = x^n + c_{n-1}x^{n-1} + ... + c_1x + c_0$ then $\alpha$ is irrational or an integer.
I honestly have no idea where to start with this, any help would be greatly appreciated.
Suppose that $\,x = a/b\,$ is a rational root, wlog in lowest terms, i.e. $\,a,b\,$ are coprime. Then
$$ 0\, =\, b^n T(a/b) = \color{#0a0}{a^n} + \color{#c00}b (c_{n-1} a^{n-1}+\cdots + c_0 b^{n-1})$$
Thus $\,\color{#c00}b\mid \color{#0a0}{a^n}.\,$ If $\,|b|>1\,$ then some prime $\,p\mid b\mid a^n,\,$ so $\,p\mid a,\,$ contra $\,a,b\,$ coprime. Therefore we conclude $\,|b| = 1,\,$ so $\,a/b\,$ is an integer if rational.
Remark $ $ This is (a special case) of half of the Rational Root Test, i.e $\,b\mid c_n\, (= 1$ here). The other half $\,a\mid c_0\,$ follows the same way (or, more slickly, by reciprocation symmetry, i.e. by applying this half to the reversed polynomial, i.e. that satisfied by $x^{-1}).$
