Let f be a polynomial of the form \(\displaystyle f(x) = x^{n} +C_{1}x^{n-1} + .....C_{n-1}x+c_{n} \)
where each coefficient \(\displaystyle c_{j} \) is an integer.
Assume that n is greater than or equal to 1, and let \(\displaystyle \alpha \) be a rational number such that \(\displaystyle f(\alpha) = 0 \)
prove that \(\displaystyle \alpha \) must in fact be an integer.
I really dont know how to even start, my teacher gives me hints on my paper,
such as
suppose \(\displaystyle \alpha = a / b \) where gcd(a,b)=1 . suppose b does not equal plus or minus 1 for a contradiction; consider prime factors of b
But I still dont know how to proceed, any help appreciated.
where each coefficient \(\displaystyle c_{j} \) is an integer.
Assume that n is greater than or equal to 1, and let \(\displaystyle \alpha \) be a rational number such that \(\displaystyle f(\alpha) = 0 \)
prove that \(\displaystyle \alpha \) must in fact be an integer.
I really dont know how to even start, my teacher gives me hints on my paper,
such as
suppose \(\displaystyle \alpha = a / b \) where gcd(a,b)=1 . suppose b does not equal plus or minus 1 for a contradiction; consider prime factors of b
But I still dont know how to proceed, any help appreciated.