To prove $f=x^4+x^3+x^2+x^1+1$ is irreducible over the $\mathbb{Q}$

Question

My line of proof is as follows:

• $\pm1$ are the only candidates for being rational roots (Rational Root Theorem)
• Since $f(1)=5$ and $f(-1)=1$, none is a root

And hence the given polynomial is irreducible over the rationals $\mathbb{Q}$.

Is this proof correct? I have doubt because the solution given in the book uses Eisenstein's criterion by modifying the polynomial to $f(x+1)=\frac{(x+1)^5-1}{x}$ and so on... which seems complicated to me.

Answer 1

(Alternative proof, not using Eisenstein's criterion.)

The polynomial is reciprocal and can be easily factored over the reals:

$$ \begin{align} x^4+x^3+x^2+x+1 &= x^2 \left(\left(x^2+\frac{1}{x^2}\right)+\left(x+\frac{1}{x}\right)+1\right) \\ &= x^2 \left(\left(x+\frac{1}{x}\right)^2+\left(x+\frac{1}{x}\right)-1\right) \\ &= x^2\left(x+\frac{1}{x}-\frac{-1+\sqrt{5}}{2}\right)\left(x+\frac{1}{x}-\frac{-1-\sqrt{5}}{2}\right) \\ &= \left(x^2+\frac{1-\sqrt{5}}{2}x + 1\right)\left(x^2+\frac{1+\sqrt{5}}{2}x + 1\right) \end{align} $$

Since neither quadratic has real roots this is the unique irreducible factorization over $\mathbb R$, and since the coefficients are not rational the polynomial is irreducible over $\mathbb Q$.

Answer 2

$f$ having no real roots doesn't mean that $f$ is irreducible over $\mathbb{Q}$, take $(X^2+1)^2$ for instance. Notice that $f=\Phi_5$ is a cyclotomic polynomial therefore it is irreducible. A way to prove it is to use Eisenstein' criterion (https://en.wikipedia.org/wiki/Eisenstein%27s_criterion) : $$f(X+1)=\frac{(X+1)^5-1}{X}=X^4+5X^3+10X^2+10X+5$$ $5$ is a prime number that divides all the coefficients of $f(X+1)$ except the leading one, and $5^2$ doesn't divide the constant coefficient squared, therefore $f(X+1)$ is irreducible over $\mathbb{Q}$, so $f$ too.