An Irreducible polynomial in $\mathbb{Q}[x]$

Question

I want to prove that $x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Q}[x]$. I noticed that this polynomial can be rewritten as $x(x+1)(x^2+1)+1$. As seen, it has no proper divisor because of the constant term $1$. However, I need a rigorous proof. Can anybody help me?

Thanks in advance.

Answer 1

This is the cyclotomic polynomial for $\mathbb{Q}(\zeta_5)$.

Let me prove a more general case:

Proposition: If $p$ is a prime number, then the polynomial $f(x) = x^{p - 1} + x^{p - 2} + \cdots + x + 1$ is irreducible over $\mathbb{Q}$.

Proof: It suffices to show that $g(x) = f(x + 1)$ is irreducible.

Since $f(x) = \frac{x^p - 1}{x - 1}$, we have $$g(x) = \frac{(x + 1)^p - 1}{x} = x^{p- 1} + \binom{p}{p - 1}x^{p-2} + \cdots + \binom{p}{2}x + \binom{p}{1}.$$ Notice that

• all the coefficients of $g$ are divisible by $p$, except the leading coefficient $1$;
• the constant term is $p$, which is not divisible by $p^2$.

It then follows from Eisenstein's criterion that $g$ is irreducible over $\mathbb{Q}$.