Let \(S\) be a set of non-zero polynomials over a field \(F\). If no two elements of \(S\) have the same degree, show that \(S\) is an independent set in \(F[x]\).
Solution
Take a finite linear relation \[ c_1f_1+\cdots+c_rf_r=0 \] with distinct \(f_i\in S\). Since their degrees are pairwise distinct, choose \(f_k\) of largest degree. The coefficient of the highest-degree term in the sum comes only from \(c_kf_k\), so it can vanish only if \(c_k=0\). Removing that term and repeating the argument shows successively that every \(c_i=0\).
Thus every finite subset of \(S\) is linearly independent, so \(S\) is an independent set in \(F[x]\).