Is $\mathbb{F}_5[x]/(x^3+2x+2)$ a field?

Question

Is $\mathbb{F}_5[x]/(x^3+2x+2)$ a field?

Since $x^3+2x+2$ is reducible over $\mathbb{F}_5$ (it has a root in $\mathbb{F}_5$), I don't no any other way to test this...

Answer 1

Hint: if $x^{3}+2x+2$ is reducible, then $\mathbb{F}_{5}[x]/(x^{3}+2x+2)$ has zero divisors. Can a field have zero divisors?

Answer 2

In general, if $f \in K[x]$ can be factored as $f=gh$ with $g,h$ not constant and coprime, then $$ \frac{K[x]}{(f)} \cong \frac{K[x]}{(g)} \times \frac{K[x]}{(h)} $$ and so cannot be a field because a product of two rings is never a field: $(1,0) \cdot (0,1) = (0,0)$.