The trace of finite dimensional extension $F$ over the finite field $K$ is surjective.

Let $K$ be a finite field, and $F$ be the finite dimensional extension field over $K$. Prove that the trace map $\operatorname{tr}_K^F: F\to K$ is surjective.

I consider the problem as follows.

Since $F$ is finite dimensional over finite field $K$, $\operatorname{Aut}_KF$ is finite and cyclic. Suppose that Aut$_KF=\langle \sigma \rangle$ of order $n$. Then $\operatorname{tr}K^F(u)=u+\sigma(u)+\cdots+\sigma^{n-1}(u)$ and $\operatorname{tr}_K^F$ is $K$-linear. We know that $\operatorname{tr}_K^F(u)\in K$, $\forall u\in F$, $\operatorname{tr}_K^F$ is not trivial, and $K$ is a 1-dimensional vector space over $K$, $\operatorname{tr}_K^F$ is surjective.

I don't know the last step is right. If it is right, is the proposition still true for any field of characteristic $p\ne 0$? Can anyone give me help? Thank you.