Here $E/k$ is a field extension. And $z$'s are elements in $E$. The subfield defined by adjoining $z$'s to $k$ is defined in the Rotman’s Advanced Modern Algebra as follow:
but it doesn’t give a concrete form of elements in $k(A)$, which is easier to understand and use. (For example, the group generated via a set has words as elements.)
Why don’t we give a form of elements in $k(A)$ too? for example, as the form: $f(x_1, x_2,\dots, x_n)/g(y_1, ..., y_m)$, where $f$ and $g$ are polynomials, and the fractal is evaluated at some finite elements of A, such that the denominator is nonzero.
Is there anything subtle I need to pay attention to?
Because a lot of the time we don't need this specific form.
But no, you are correct : they all have this form; and it's an easy exercise to prove that they do :
a) Prove that the elements of this form are a subfield of $K$; that it contains $k$ and $A$.
b) Prove that any subfield of $K$ containing $k$ and $A$ contains all elements of this form.
c) Conclude.