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Let $f(X)$ be a polynomial with integer coefficients such that $f : \mathbb{Z} \longrightarrow \mathbb{Z}$ is onto. Show that $f(X)=\pm X+c$ for some integer $c$.

I was given this question in a lecture on irreducible polynomials. I think a solution using Ostrowski's criterion was presented, but I couldn't quite follow every step. Could someone help? Thanks

doingmath
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  • My thoughts would be: if $f$ is even degree, it has either a minimum or maximum value, so it can't be onto. If $f$ has odd degree $\ge 3$, then $\Delta f(x) \to \infty$ as $x \to \pm \infty$ or $\Delta f(x) \to -\infty$ as $x \to \pm \infty$, so there's no way $f$ can be onto. So $f$ has degree 1, and from there it's easy to conclude the coefficient of $x^1$ is $\pm 1$. – Daniel Schepler Jul 12 '17 at 23:47

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Hint: For $x \gg 1$, we have that $f(x) \approx x^{\deg f}$. For $f: \mathbb{Z} \to \mathbb{Z}$, this can't possibly be onto.

cactus314
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