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Let $f \in \mathbb{Z}[x]$ be monic, irreducible and hyperbolic (no roots of absolute value $1$), and such that $f(0)= \pm 1$. Denoting as $c_{p}(x)$ the cyclotomic polynomial $$c_{p}(x)=1+x+\cdots +x^{p-1},$$ my question is: how can be characterized the (certainly finite?) set of primes $p$ for which $f$ and $c_{p}$ are co-prime, ie there are polynomials $u$ and $v$ in $\mathbb{Z}[x]$ such that $$c_{p} u+f v=1.$$ As an obvious consequence, we have that if $\zeta$ is a primitive $p$-root of unit, $f(\zeta)$ is a unit in $\mathbb{Z}[\zeta]$ and so, for each $1\leq k\leq p-1$, $$f(\zeta^{k})=\zeta^{j} r$$ for some $j$, where $r$ is a real unit of that ring.

Pedro Martins Rodrigues
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1 Answers1

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Not an answer, but the set of $p$ that you define is quite likely infinite (Oops, see David's comment). Here is an argument. Let $K$ be the splitting field of the polynomial $f(x^2)$ and $\ell$ a prime that splits completely in $K$. There are lots of those primes and is a reasonable assumption (?) that infinitely many of them satisfy $\ell - 1 =2p$, $p$ prime. Given such an $\ell$ and a root $\alpha$ of $f \mod \ell$, by construction, $\alpha$ is a square, so $\alpha^p = \alpha^{(\ell-1)/2} = 1$ and $\alpha$ is a common root of $f$ and $c_p$ modulo $\ell$ and $p$ is not in your set.

A better argument follows. The resultant of $f$ and $c_p$ grows like $a^{p-1}$ where $a$ is the Mahler measure of $f$, which is $>1$ under the hypotheses. So indeed the set of primes you define is finite.

Felipe Voloch
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  • Felipe, I'm confused. You argue that a lot of primes are NOT in the set, and then conclude from this that the set is probably infinite. In any case, it is certainly true that there are infinitely many $\ell$ such that $f(x^2)$ splits mod $\ell$ and $\ell \equiv 1 \mod 2p$ -- just apply Cebatarov to the compositum of the $2p$-th cyclotomic field and the splitting field of $f$. – David E Speyer Sep 23 '11 at 12:31
  • Thank you very much. The idea of computing the resultant had occurred to me but I was unaware of the link with the Mahler measure. Can you indicate a nice reference for it? I thought that it should be possible to determine/characterize those primes in terms of the arithmetic of $\mathbb{Z}[x]/(f)$. – Pedro Martins Rodrigues Sep 23 '11 at 14:52
  • Reference: Lehmer, D. H. Factorization of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), no. 3, 461–479. – Felipe Voloch Sep 23 '11 at 16:30
  • I found an interesting reference to this:

    Kaminski, M., "Cyclotomic Polynomials and Units in Cyclotomic Number Fields", J. of Number Theory, 28, 283-287, (1988)

    The author proves that if $P(x)$ is a monic, irreducible polynomial with integer coeficients ($P(x) \neq x$) and $P(e^{2 \pi i/n})$ is a unit in the ring $\mathbb{Z}[e^{2 \pi i/n}]$ for infinitely many $n$, then $P$ is cyclotomic itself.

     – Pedro Martins Rodrigues Sep 27 '11 at 12:32