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I need to show that for any $n$, I need to show that there is an irreducible polynomial in $Q[x]$ of degree $n$ having exactly $n-2$ real roots. As a hint I have that if $f(x) \in \mathbb{R}[x]$ is a any polynomial having exactly $k$ distinct real roots, there exists $\epsilon > 0$ for which $f(x) +a$ has exactly $k$ real roots, for all $a\in \mathbb{R}$ with $|a|<\epsilon$.

Then by starting with any polynomial $f(x) \in Q[x]$ with exactly $n􀀀-2$ distinct real roots, and using paragraph above $f(x)+a$ has the same property for infinitely many $a\in Q$. Now, make a judicious choice of $f(x) \in Z[x]$ and $a \in Q$ for which the Eisenstein irreducibility criterion can be applied.

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If I understand the hints correctly, you are asked to explore $$ f(x)=(x-1)(x-2)...(x-(n-2))·(x^2+1)+ε $$ or, perhaps more suited for the modular explorations, $$ f(x)=(x^2+1)·(x-2)(x-4)(x-8)...(x-2^{n-2})+ε $$

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