TITLE: Check the proof of:$x^2+1$ is irreducible over $\Bbb Q$
QUESTION [14 upvotes]: Prove: $x^2+1$ is irreducible over $\mathbb{Q}$
Proof: Since $x^2+1=(x-i)(x+i)$, so $x^2+1$ is irreducible over $\mathbb{Q}$. 
Is it right?

REPLY [11 votes]: The polynomial $f(x)$ is irreducible if and only if $f(x+1)$ is irreducible. But in your case,
$$f(x+1) = (x+1)^2 +1 = x^2 + 2x + 2$$
is irreducible by Eisenstein's criterion (with $p=2$.)