Suppose $L/K$ and $M/K$ are algebraic extensions of a field $K$, such that $L\cap M=K$, and $L/K$ is a normal extension. It is well-known that, with these conditions, we have:

$$\text{Gal}(L/K)\cong \text{Gal}(LM/M).$$

However, suppose we now complicate matters by specifying that $M/K$ (and hence also $LM/L$) is not algebraic but transcendental ($L/K$ remaining normal). In this case, am I right in thinking that this identity still holds?

If not, can anyone provide a counterexample, or is there an obvious reason why this doesn't work? What if we specify that $K$ has characteristic zero?