$K$ is an infinite field of characteristic $p\gt0$. Prove that the multiplicative group $K^{\times}=K\setminus\{0\}$ is not cyclic.
I want to consider this problem by two cases. One is to prove that it cannot be true when $K^{\times}=\langle \alpha \rangle$ where $\alpha$ is algebraic over $\mathbb Z_p$ which is viewed as the prime field of $K$. With people's help, I have understood this case.
However, I have no idea how to prove that it is also not true that $K^{\times}=\langle \alpha \rangle$ when $\alpha$ is transcendental over $\mathbb Z_p$.
Any help is appreciated.