I'm trying to wrap my head around a proof in my lecturer's Galois theory notes, but am struggling to see why a particular step follows from what came before it. I've attached a screenshot to save me copying out the whole proof (it's quite long, and it's only a very small part I'm having trouble with).
I'm struggling to see how the very last step follows: I can't see how from (3) you can can deduce that $\sum \alpha_i\frac{\beta_i}{\beta_1} = 0$ Presumably you take $g_j$ to be the identity so that you get $\alpha_i$ out, but when you substitute in $g(\beta_i) = \frac{\beta_i}{\beta_1}g(\beta_1)$ you're left with a $g(\beta_1)$ that doesn't feature in the given sum.
I'd really appreciate if someone could explain what I'm missing or, if there's some kind of mistake in the proof, what I could do to fix it.