TITLE: Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\,\left(\Bbb Z/3\Bbb Z\right)[x]\;$......
QUESTION [2 upvotes]: I am stuck on the following problem:  

Let $f(x)=x^3+2x^2+1,\,\,g(x)=2x^2+x+2.$ Then over $\,\left(\Bbb Z/3\Bbb Z\right)[x]\;$, show that $f(x)$ is irreducible ,but $g(x)$ is not.

Can someone explain how to tackle it? Thanks in advance for your time.

REPLY [3 votes]: Hints:
== Over any field, a polynomial of degree $\,\le 3\;$ is irreducible iff it has not root in that field
== $\,1^2-4\cdot2\cdot2=-15=0^2\pmod 3\;$, and thus in fact $\,g\,$ is a perfect square in $\,\left(\Bbb Z/3\Bbb Z\right)[x]\;$

REPLY [1 votes]: Hint: What happens if these polynomials have roots in $\mathbb Z_3$? What possible roots can they have? Can you check them?
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