I am wondering how to see whether or not $f(x) = x^4 – 6x^2 + 3x + 57$ is irreducible over $\mathbb{C}$ and $\mathbb{R}$ ,respectively.
One may start by finding a rational root. But the polynomial is $3$-Eisenstein, so it is irreducible over $\mathbb{Q}$.
One can try to factor the polynomial by a quick-witted, algebraic trick, but I'm not seeing how one can do this with $f(x)$.
There is a formula for roots of a quartic polynomial, but it is very complicated, and hard to commit to memory.
Is there an easier way to see whether or not $f$ is irreducible over $\mathbb{C}$ and $\mathbb{R}$ ? I suspect it is irreducible over both, but how can I see this?
Thanks!
Best Answer
The answer is no and no.
Only polynomials of degree $1$ are irreducible over $\mathbb C$.
Only polynomials of degree $1$ or $2$ are irreducible over $\mathbb R$.
These statements are equivalent to the fundamental theorem of algebra.