How can I show that $x+1$ and $x^2+x+1$ are irreducible in $\mathbb{R}[x]$?

Question

How can I show that $x+1$ and $x^2+x+1$ are irreducible in $\mathbb{R}[x]$?

For $x+1$, I'm not sure if it suffices to say $x+1$ has degree 1 so it is irreducible?

Answer 1

Yes, for $x+1$, as you cannot factor it further! Now notice for the degree $2$ polynomial, it is irreducible if and only if it has a root . Does $x^{2}+x+1$ have a root in $\mathbb{R}$?

Answer 2

As for your 1st question:

Since $x+1$ cannot be expressed as a product of $g(x)h(x)$ where $g(x),h(x)\in \mathbb{R[x]}$ both of lower degree than the degree of $x+1$. Hence, $x+1$ is irreducible over $\mathbb{R}$.

Now, for your 2nd question:

Since $x^2+x+1$ has no zeros in $\mathbb{R}$. Thus $x^2+x+1$ is irreducible over $\mathbb{R}$, for factorization $x^2+x+1=(ax+b)(cx+d)$ for $a,b,c,d\in \mathbb{R}$ would give rise to zeros of $x^2+x+1$ in $\mathbb{R}$.

However, $x^2+x+1$ is not irreducible over $\mathbb{C}$, because $x^2+x+1$ factors in $\mathbb{C}[x]$ into $(x+\frac{1+\sqrt3i}{2})(x+\frac{1-\sqrt3i}{2})$.