Use the rational root theorem.
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.
Use the rational root theorem.
suppose
with p,q integers with gcd(p,q) = 1.
then
.
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.