I have a question about an argument used in Szamuely's "Galois Groups and Fundamental Groups" in the excerpt below (or look up at pages 73/74):
In the proof we construct locally in $U$ a polynomial
$$F= \prod (t-f_i) =t^d +a_{n-1}t^{d-1} + ... + a_0 \in \mathcal{M}(U)[t]$$
with $a_j$ as elementary symmetric polynomials of the $f_i$ are moromorphic on $U$.
In order to glue these $F$ together to a polynomial in $\mathcal{M}(X)[t]$ we must check that the $a_i$ on the overlaps of two open sets $U, W$ coinside (then we can "glue" $F_U,F_W$ together).
I don't understand why the $F$ (resp the a_i$'s) must coinside on the overlaps. Does it depend on a choice, doesn't it?