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

Use the rational root theorem.

Thank you for the starting point!!!

suppose

with p,q integers with gcd(p,q) = 1.

then

.

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.