In my abstract algebra class, we were given some polynomials to prove irreducible or not over the rationals. I have come up with a proof for this one, but I'm not sure if it's valid.
The polynomial: $$x^4 + 2x^3 + x^2 + x + 1$$
I've started by taking the modulus $2$ of all the coefficients to get:
$$x^4 + x^2 + x + 1$$There's a theorem in my book that says if you take the modulus using a prime and the new polynomial has the same degree and is irreducible, than the previous polynomial is also irreducible.
So to prove $x^4 + x^2 + x + 1$ is irreducible:
If it is reducible it can be written as $f(x)g(x)$. But because it has a constant term, both $f$ and $g$ must have a constant term. But this also implies neither $f$ nor $g$ can have an $x^3$ term because their product does not have an $x^3$ term. But since the product has an $x^4$ term, they both must have $x^2$ terms. Also, since the product has an $x$ term, at least one of $f$ and $g$ must have an $x$ term. But then the $x$ term times the $x^2$ term in the other would make an $x^3$ term, which the product does not have, so $f$ and $g$ cannot exist.
Is this proof solid? Is there a more elegant way?