- #106

julian

Gold Member

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I meant to write ##f(x) - f(a)## on the LHS. I was just simply stating an identity, I make appropriate replacements for ##x## and ##a## in the next part.The ##a## on the right-hand side should be an ##\alpha .##

Typo.The first equality sign has to be less or equal.

I thought I had to consider ##\tilde{\alpha} \in \{ r_1 , \dots , r_k \}## separately as the proof used in case (a) used that ##f(\tilde{\alpha}) \not= 0##. But of course I didn't have to do this because, as you alluded to, if ##p/q## were a root then you could factor out ##x - p/q## from ##f (x)## and ##\alpha## would satisfy a polynomial with rational coefficients whose degree is less than ##n##.I assume this is meant to be the other way around since we already covered all cases of ##\alpha \not\in \{ r_1 , \dots , r_k \}.##

The good news is that it is irrelevant because we may assume ##f(x)## to be irreducible over ##\mathbb{Q}##.

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