According to Wikipedia, normal extension are assumed to be algebraic. But one of the definitions
$K/k$ is normal if any $k$-embedding $\sigma : K \rightarrow \Omega$ of $K$ into a fixed algebraic closure $\Omega$ of $K$ does not leave $K$ i.e. $\sigma(K) = K$
is applies perfectly to any field. What happens to transcendental extension $K/k$ that satisfies this property? Are they all essentially algebraically closed?
EDIT: Thanks to @Joanpemo, I realize that I made a mistake in the definition of normal extension: $\Omega$ is supposed to be algebraic closure of the smaller field $k$, not of $K$. Nevertheless, the question is still applicable: We can still ask for transcendental extension $K/k$ satisfying my wrong definition.