Prove or disprove that the quotient ring is a field: $\displaystyle \frac{\mathbb Z_5[x]}{\langle 4x^3+ x^2+3\rangle}$ Prove or disprove that the quotient ring is a field: $$\frac{\mathbb Z_5[x]}{\langle 4x^3+ x^2+3\rangle}$$
Okay, so I need to either find an element that doesn't have an inverse or prove that they all have inverses, correct?
IF it is a ring, how can I prove it without checking every element in the ring?
 A: Hint:
use the fact that a quotient ring $ R / I$ is a field if and only if $ I  $ is a maximal ideal
A: The polynomial occured in the question is not irreducible, because it has $x-3$ as a factor. So the qoutient ring is not field but is a ring because qoutient of a ring on an ideal is a ring.
A: It's obvious $f$'s coef's have alternating sum $\,3\!-\!0\!+\!1\!-\!4 = 0\,$ so $\,f(-1)= 0\,$ so $\,f(x) = (x\!+\!1)g\,$ therefore ${\rm mod}\ f\!:\  f = (x\!+\!1)g\equiv 0\,$ but $\,x\!+\!1\not\equiv 0,\, g\not \equiv 0\,$ so $\,\Bbb Z_5[x]/f\,$ is not a field.
Remark $\ $ Since the nontrivial factorization of $\,f(x)\,$ in $\,\Bbb Z[x]\,$ persists in every nontrivial ring $R,\,$ the result holds true more generally for all such quotient rings $\,R[x]/f$.
A: Hint: Check if $f(x) = 4x^3 + x^2  +3$ is irreducible over $\mathbb Z_5$. And use that $$\frac{\mathbb Z_5 [x]}{\langle 4x^3  +x^2 + 3\rangle}$$
is a field, if and only if, $f$ is irreducible over $\mathbb Z_5$. Notice that $\overline {3}$ is a root of $f$. 
