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

• Sep 18th 2012, 09:47 AM
bskcase98
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.
• Sep 18th 2012, 10:42 AM
emakarov
Re: need help with a proof if f(b) = 0 and b is not an integer, then b is irrational
• Sep 18th 2012, 04:32 PM
bskcase98
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!!!
• Sep 18th 2012, 10:17 PM
Deveno
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 pn, but gcd(p,q) = 1, so gcd(pn,q) = 1.

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