# Relationship between cyclotomic polynomials

Let $n, m$ be two natural numbers and $\Phi_n(q), \Phi_m(q)$ the $n$-th and $m$-th cyclotomic polynomials respectively. Define a function $c_{n,m} \colon \mathbb{N} \to \left\{0,1\right\}\cup \left\{ p \mid p \, \text{prime}\right\}$ by $$c_{m,n} = \begin{cases} 0, \, & \text{if n = m}\\ p, \, & \text{if \frac{n}{m} = p^j where p prime and j \in \mathbb{Z}\setminus \{0\}}\\ 1, \, & \text{if \frac{n}{m} is not an integer power of a prime} \end{cases}$$

Show that $\Phi_m(q) \in \sqrt{\left( \Phi_n(q)\right) + (c_{m,n})[q]}$

This result should rely on properties of cyclotomic polynomials I am not aware of. Obviously if $c_{n,m} = 0$ the result is true since $n = m$ and $(c_{n,m}) = (0)$. If $c_{n,m} = 1$, then $(c_{m,n}) = (1) = \mathbb{Z}$, and then it is true that $\Phi_m(q) \in \sqrt{\left( \Phi_n(q)\right) +\mathbb{Z}[q]}$, correct?

Any answer or any reference to literature where this results or proven would be appreciated. Thank you in advance.

• Your notation is really unclear. Using the $\in$ symbol seems to indicate that you intend $\sqrt{\left( \Phi_n(q)\right) + (c_{m,n})[q]}$ to denote a set; what is the exact definition of this set? – Greg Martin Jul 14 '18 at 18:03
• @GregMartin The notation $\sqrt{(\Phi_n(q))+ (c_{m,n})[q]}$ means the radical of $(\Phi_n(q))+ (c_{m,n})[q]$. I hope it is clear now. – user313212 Jul 14 '18 at 18:39

As you write, the statement in case $c_{m,n} = 1$ is immediate because clearly $\Phi_m(x) \in \mathbb{Z}[x] = \mathbb{Z}[x] + (\Phi_n(x)) = \sqrt{\mathbb{Z}[x] + (\Phi_n(x))}$. Since this is true for all $m,n$, the statement in this case is also not meaningful.
If $n = m p^i$ so that $c_{m,n} = p$, then we have $\Phi_n(x) = \Phi_m(x^{p^i})$ (elementary property of cyclotomic polynomials).
Therefore it suffices to prove that $$\Phi_m(x)^{p^i} \equiv \Phi_m(x^{p^i}) \mod p$$
But in fact this statement is more generally true of any polynomials over $\mathbb{Z}/p\mathbb{Z}$. It can be seen as an elaboration of the so-called "Freshman's Dream," that $(x+y)^{p} \equiv x^p + y^p \mod p$. I also found a reference here as lemma 4.3. The first property I mentioned is theorem 3.2 there.
• Thank you! The case where $c_{m,n} = 1$ is okay as I wrote on the question? – user313212 Jul 15 '18 at 11:01