I am trying to prove the Cyclotomic polynomial is irreducible over $\mathbb{Q}[x]$ for any prime $p$ using Eisenstein's Criterion. However, I would like to be more specific and prove the following Lemma using induction on $n$. I understand the base case, but I just need a little help with the induction hypothesis.
Lemma: Let $R$ be a commutative ring with identity $1$. Let $(p)$, where $p$ is a prime number, be a prime ideal, and $a_1, \ldots, a_n \in \mathbb{R}$. If $a_1\cdots a_n \in (p)$, then $a_i \in (p)$, for some $i \in \{1, \ldots, n\}$.
Please refrain from proving the polynomial is irreducible, as I would like to do it myself.
Thank you for your time.
Suppose $R$ is a commutative ring with a prime ideal $\mathfrak p$, and suppose $a_1\cdots a_n \in \mathfrak p$. Since $\mathfrak p$ is prime, either $a_n \in \mathfrak p$ or $a_1 \cdots a_{n-1} \in \mathfrak p$. Now use the induction hypothesis.