Given a polynomial over $Z$, $P(X)$, I'd like to show that for any $i$ in $N$, there is a whole number $r$ so that $P(x)+r*x^i$ is irreducible.
I'd preferably would love a elementary argument so that I can understand it.
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2This is a special case of Hilbert's Irreducibility Theorem. Even for this special case there might not be a proof that's both elementary and short enough for Stackexchange. https://en.wikipedia.org/wiki/Hilbert%27s_irreducibility_theorem – Noam D. Elkies May 26 '16 at 00:53
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@NoamD.Elkies Thanks! My lecturer quoted this theorem when I asked the question. I was hoping there is an elementary argument (which is the case for $i=0$ for example). I guess I'll buckle up and look for the proof of Hilbert's Irreducibility Theorem. – May 26 '16 at 00:56