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The question is

Show that $f(n)=n^5+n^4+1$ is not prime for $n>4$.

The solution is given as

Let $\omega$ be the third root of unity. Then $\omega^2+\omega+1=0$. Since $\omega^5+\omega^4+1=\omega^2+\omega+1$, we see that $\omega^2+\omega+1$ is a *factor of the polynomial. So *$n^2+n+1|n^5+n^4+1$.

Which polynomial are we referring to in the bold typeface above? And how is $n^2+n+1|n^5+n^4+1$ true due to $\omega^5+\omega^4+1=\omega^2+\omega+1$?

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If two polynomials $f(x)$ and $g(x)$ have a common factor, it has a common root. In this case, all of the roots of $g$ are in $f$. –  Balarka Sen Dec 27 '13 at 18:12
    
What is $g(x)$ here and what are the common factors? –  user117913 Dec 27 '13 at 18:15
    
$g(x)$ is $x^2+x+1$ and the common factors are $x = 1, w, w^2$. –  Balarka Sen Dec 27 '13 at 18:20

4 Answers 4

up vote 1 down vote accepted

This is rather poor phrasing (in the highlighted solution). What it is saying is that that $\omega,$ which is a root of the irreducible polynomial $x^2+x+1$ is also a root of $x^5+x^4+1,$ and so the gcd of the two polynomials is not $1,$ and since the first polynomial is irreducible, it must divide the second. Now, that said, you can verify this without any fanciness by long division.

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As the 3rd root of identity $\omega=\mathrm{e}^{2\pi i/3}$ satisfies $x^5+x^4+1=0$, then so does its conjugate $\bar\omega=\mathrm{e}^{-2\pi i/3}$, and hence $(x-\omega)(x-\bar\omega)=x^2+x+1$ divides $x^5+x^4+1$.

Note that $$ x^5+x^4+1=(x^2+x+1)(x^3-x+1). $$

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I know other solutions. Please help me understand the given one. –  user117913 Dec 27 '13 at 18:16
    
The given solution tells you how to get this factorization. –  Igor Rivin Dec 27 '13 at 18:20
    
See edited version. –  Yiorgos S. Smyrlis Dec 27 '13 at 18:23
    
This is the solution that I was looking for. –  user117913 Dec 27 '13 at 18:28

See that $\omega$ is a root of both $f(x)=x^5+x^4+1$ and $g(x)=x^2+x+1$. Now $g(x)$ has all coefficients real, and it is a polynomial of degree 2. Hence it has two roots and the second root must be the complex conjugate of $\omega$, that is $\omega^2$. Hence $g(x)=(x-\omega)(x-\omega^2)$. Also note that $f(\omega^2)=0$. Hence $f(x)=(x-\omega)(x-\omega^2)h(x)=g(x)h(x)$ for some real polynomial $h(x)$. Now you can easily show by equating coefficients that $h(x)$ has integer coefficients. Now put $x=n$ on both sides. Then since $h(n)$ turns out to be an integer $n^2+n+1|n^5+n^4+1$.

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Let $$g(n)=n^2+n+1$$ and $$h(n)=n^3-n+1$$.

Clearly $$f(n)=g(n) \cdot h(n)$$

Now $h(1)=1$ and $h'(n) = 3 n^2-1 > 0$ for $n\ge 1$. So, for $n>1$ both $g(n)$ and $h(n)$ are greater than $1$. So

$f(n)$ **is composite for all $n>1$

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