Question 73
Let $n\ge2$ be a positive integer, let $(K,+,\cdot)$ be a commutative field satisfying $\underbrace{1+\cdots+1}_{m\text{ times}}\ne0$ for $m=2,\ldots,n$, let $f\in K[X]$ have degree $n$, and let $G\ne K$ be a subgroup of the additive group $(K,+)$. Show that there exists $a\in K$ such that $f(a)\notin G$.