Wednesday, 20 April 2016

abstract algebra - On divisors of pn1

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(xλi1) such that deg(f)>n and f(p)pn1. For example, take p=3, n=2 and f(x)=(x1)3 we have (311)(311)(311)321. However, with assumptions p3 and n4, I could not find any such polynomial f with deg(f)>n and f(p)pn1. So it made me to claim that the following statement might be true:




Let p3 be a prime number and n4. Then for every f(x)Z[x] of the form f(x)=mi=1(xλi1) such that deg(f)>n, we have f(p)pn1.




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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