Given a field $K$ and infinite set $B$. Suppose that we have the transcendental degree of $K(B)$ over $K$ is $|B|$, i.e $[K(B):K]_t=|B|$. Can we conclude that $B$ is transcendental basic of $K(B)$ over $K$?
If $B$ is finite, then clearly $B$ is a transcendental basic. For $B$ is infinite, I think there are still possibilities that there exists a proper subset $C$ of $B$ such that $C$ is transcendental basic and $|C|=|B|$. In other words, there could be that $B$ is not algebraically independent, but I haven't figured out how to construct such an extension. My question is if $B$ is infinite, can we conclude that $B$ is transcendental basic? If not, how I can construct a counterexample?
Any helps would be appreciated! Thanks