need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

I do not even know where to start with this. this proof is for an abstract algebra course.

Suppose f(x) = x^{n }+ a_{1}x^{n-1} + . . . + a_{1}x + a_{0} where the coefficients a_{0},a_{1}, . . . ,a_{n-1} are integers.

prove: if f(b) = 0 and b is not an integer, then b is irrational.

Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

Thank you for the starting point!!!

Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational

suppose

$\displaystyle f\left(\frac{p}{q}\right) = 0$ with p,q integers with gcd(p,q) = 1.

then

$\displaystyle p^n = q(-a_{n-1}p^{n-1}-\dots-q^{n-2}a_1p-q^{n-1}a_0)$.

then q divides p^{n}, but gcd(p,q) = 1, so gcd(p^{n},q) = 1.

these two facts together imply q = 1, so p/q is an integer.