For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

Question

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

What I tried: To show that $f$ is not a unit I did the following. Suppose that $f$ is a unit, then there exists a polynomial $g(x)\in K[x]$ such that $f(x)g(x) = 1$. But the left hand side has degree greater or equal to four, which leads to a contradiction.

Then to show that $f$ is irreducible I tried several things (to factor $f$). First I substituted $(1-x)$ for $x$ to see if I could then nicely factor the polynomial but no luck. I also substituted $y = x^2$ and tried to factor the polynomial $y^2+x +1$, but again, no luck. Thanks for any help.

Answer 1

$x^4+x^2+1=(x^2+1)^2-x^2=(x^2+x+1)(x^2-x+1)$