This question raised from discussions around my previous question. This may seem trivial or easy, but I am so confused and can't see the answer. So I will be so grateful if you would help me please.
For the natural number n and prime number p, it is likely possible that there exists a polynomial f(x)∈Z[x] of the form f(x)=∏mi=1(xλi−1) such that deg(f)>n and f(p)∣pn−1. For example, take p=3, n=2 and f(x)=(x−1)3 we have (31−1)(31−1)(31−1)∣32−1. However, with assumptions p≥3 and n≥4, I could not find any such polynomial f with deg(f)>n and f(p)∣pn−1. So it made me to claim that the following statement might be true:
Let p≥3 be a prime number and n≥4. Then for every f(x)∈Z[x] of the form f(x)=∏mi=1(xλi−1) such that deg(f)>n, we have f(p)∤pn−1.
Is the above assertion true? If yes, would you please hint me how to prove it (or refer me to a reference)?
Thank you in advance.
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