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

1. ## 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) = xn + a1xn-1 + . . . + a1x + a0 where the coefficients a0,a1, . . . ,an-1 are integers.
prove: if f(b) = 0 and b is not an integer, then b is irrational.

3. ## 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!!!

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

suppose

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

then

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

then q divides pn, but gcd(p,q) = 1, so gcd(pn,q) = 1.

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