Another Number Theory Question about Polynomial - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T22:58:36Z http://mathoverflow.net/feeds/question/110915 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110915/another-number-theory-question-about-polynomial Another Number Theory Question about Polynomial Lwins.Gafield 2012-10-28T16:38:06Z 2012-10-28T18:27:48Z <p>We know following theorem by Schur: Suppose that $f(x) \in \mathbb{Z}[x]$ is a polynomial such that exists an integer $m$ such that $f(n) = m^2$ for every integer $n$. Then $f(x) = g(x)^2$ for some $g(x) \in \mathbb{Z}[x]$.</p> <p>Is this true if I using $3,4,5 \cdots$ instead of $2$?</p> http://mathoverflow.net/questions/110915/another-number-theory-question-about-polynomial/110928#110928 Answer by Peter Mueller for Another Number Theory Question about Polynomial Peter Mueller 2012-10-28T18:25:13Z 2012-10-28T18:25:13Z <p>I think the proposer tried to say that <em>if for every integer $n$ there is an integer $m$ such $f(n)=m^2$, then $f(x)=g(x)^2\dots$</em>.</p> <p>The generalization from exponent $2$ to higher exponents $e$ is an exercise in the second volume of the classical Polya/Szegö problem book. I don't have it at hand now, so I cannot give a more precise reference.</p> <p>If one allows more sophisticated tools, then it follows immediately from Hilbert's irreducibility theorem: Let $t$ be another variable, and factorize $f(x)-t^e$ over $\mathbb Q(t)$ into irreducible factors $P_i(t,x)$. By Hilbert, there are infinitely many integers $n$ such that $P_i(n,x)$ is irreducible for all $i$. On the other hand, $f(x)-n^e$ has an integral root for each $n$. Thus there is an index $i$ such that $P_i(t,x)$ has degree $1$ with respect to $x$, so $f(x)-t^e$ has a root in $\mathbb Q(t)$. Gauss' lemma yields that this root is in $\mathbb Q[t]$, so $f(t)=g(t)^e$ for a polynomial $g$ with rational coefficients. One quickly gets that the coefficients of $g$ then are indeed integers. </p> http://mathoverflow.net/questions/110915/another-number-theory-question-about-polynomial/110929#110929 Answer by Robert Israel for Another Number Theory Question about Polynomial Robert Israel 2012-10-28T18:27:48Z 2012-10-28T18:27:48Z <p>Yes. See e.g. <a href="http://www.isibang.ac.in/~sury/polys.pdf" rel="nofollow">http://www.isibang.ac.in/~sury/polys.pdf</a> Corollary 3.3.</p>