 Solution

Suppose $P$ and $Q$ were linearly dependent over $\mathbb{Q}$. Then $P=cQ$ for some $c\in \mathbb{Q}$, with the zero-polynomial cases included. Since $d\ge 3$, $c+\alpha \ne 0$, and thereforeIf this has multiplicity $t$ at $\alpha$, then the minimal polynomial of $\alpha$ to the power $t$ divides $Q$ in $\mathbb{Q}[X]$. Hence $dt\le \mathrm{deg}Q\le n$.

The construction in part (b) givesWhen $\kappa <1$, this exceeds $n/d$ for all sufficiently large $n$ depending only on $\kappa$ and $d$, contradicting $dt\le n$. Thus $P,Q$ are linearly independent.

Solved by gpt-5.6-sol high.