Prove that this isomorphism is identity on K(Hungerford Theorem 1.5)

Question

While self-studying algebra from Thomas Hungerford I got struck on theorem 1.5 of Chapter Fields and Galois Theory.

Question: I am not able to figure out how does $\phi$ is identity over K?

I think this could be due to I am wrong in assuming what " Identity over K " means. I think it means $\phi(k) $ = k' for some k'$\in$ K for all k in K.

But I can't prove it.

Kindly tell me what mistake I am making and how to rightly prove it.

Answer 1

$\phi$ being the identity on $K$ means that for each $k \in K$, $\phi(k) = k$. For example, if $k = 12345 \in K$, then $\phi(12345) = 12345$.

Your definition - the statement that there exists as $k_0 $ such that $\phi(k) = k_0$ for all $k \in K$ - is the statement that $\phi$ is constant on $K$. That's a different concept.