Prove that polynomial $1+x+x^2$ of $\mathbb{R}[x]$ is irreducible over $\mathbb{R}$. Do not use the Eisenstein criterion.

Question

Prove that $1+x+x^2$ is irreducible over $\mathbb{R}[x]$. Do not use the Eisenstein criterion.

I don’t know where to start. Could someone help me?

Answer 1

For all $x\in\mathbb R$, $x^2+x+1=\left(x+\frac12\right)^2+\frac34>0$, so $x^2+x+1$ has no roots in $\mathbb R$.

A quadratic polynomial with no roots is irreducible.