Let $f(x)\in \mathbb{Q}[x]$ be a polynomial of degree $n>0$. Let $p_1, p_2, \dots ,p_{n+1}$ be distinct prime numbers. Show that there exists a non-zero polynomial $g(x)\in \mathbb{Q}[x]$ such that $fg=\sum_{i=1}^{n+1} c_ix^{p_i}$ with every $c_i\in \mathbb{Q}$.
Asked
Active
Viewed 45 times
0
-
2Consider the images of the $x^{p_i}$ in the quotient ring $\Bbb Q[X]/(f(X))$. – Angina Seng Oct 10 '17 at 05:51
-
1Probable duplicate of https://math.stackexchange.com/questions/2232316/polynomial-rings-and-prime-numbers. – lhf Oct 10 '17 at 12:58