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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!

Sil
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michiganbiker898
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1 Answers1

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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.

lhf
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