My question refers to a step in the proof of Prop. 3.3.5 Szamuely and Tamásin's "Galois groups and fundamental groups":
Here the statement and Thm 3.3.3 & lemma 3.3.6: The main ingredients for the proof:
AND here the proof of 3.3.5 with red tagged unclear argument:
My question is why $f$ satisfies an irreducible polynomial equation $(\phi^*a_n)f^n + ... +(\phi^*a_0)=0$? Why irreducible?
Lemma 3.3.6 says that the polynomial equation is not neccessary irreducible.
Intuitively I guess that should have something to do with the assumption that all values $f(y_i)$ are distinct. Why does it suffice? I don't find a clear argument.