Question 30
Let $n \ge 2$ be an integer and let $(K, +, \cdot)$ be a commutative field with the property:
$$
\underbrace{1 + \cdots + 1}_{m \text{ times}} \neq 0, \quad m = 2, \dots, n.
$$
Consider a polynomial $f \in K[X]$ of degree $n$ and $G$ a subgroup of the additive group $(K, +)$, $G \neq K$. Prove that there exists $a \in K$, such that $f(a) \notin G$.
$$
\underbrace{1 + \cdots + 1}_{m \text{ times}} \neq 0, \quad m = 2, \dots, n.
$$
Consider a polynomial $f \in K[X]$ of degree $n$ and $G$ a subgroup of the additive group $(K, +)$, $G \neq K$. Prove that there exists $a \in K$, such that $f(a) \notin G$.