I am currently reading Chapter 3.4 of Galois group and Fundamental group by of Szamuely and I am stuck at an important step of lemma 3.4.2.
I am not sure I understand what they mean by a "direct factor" of the tensor product, and what they mean by "the component coming from one of these...we are done."
I tried working through an example: If I look at $X= \mathbb{P}^1$ the Riemann sphere, we have $M(X)=\mathbb{C}(t)$. Suppose I take $Y_1, Y_2$ to be the branched cover that maps $z \mapsto z^2$ on the restriction $\mathbb{C}^*$. If I understand correctly, the corresponding field extensions are $\mathbb{C}(t)[X]/(X^2-t)$ and $\mathbb{C}(t)[Y]/(Y^2-t)$. Then their tensor product over $\mathbb{C}(t)$ gives me $\mathbb{C}(t)[X,Y]/(X^2-t, Y^2-t)$. So if I take "a minimal polynomial of the generator" like in the proof, say $X^2-t$ into $\mathbb{C}(t)[Y]/(Y^2-t)$, I get $X^2-Y^2=(X+Y)(X-Y)$, product of two irreducible factors. But I am not sure what to do next?
Thank you in advance!