differential algebraic structure theorems

Salvete,

I am trying to get a clearer view on differential algebraic structure theorems, but it seems I am reading even the basic theorems incorrectly. Is there a kind soul around who could point out my errors?

In Manuel Bronstein's Symbolic Integration I, second edition, Corollary 9.1.1 reads:

| Let k subset K be fields of characteristic 0. Then, B subset K is
| algebraically independent over k if and only if {db | b in B}
| subset Omega_{K/k} is linearly independent over K.

So, as my first example, I used k=Q, K=Q(x,t) with dt=t*dx, i.e. t=exp(x). B={x,t} is clearly algebraically independent over k, but
{db | b in B} = {dx, t*dx} is obviously not linearly independent over K.


Regards,
Christopher
.