Are $\mathbb{Z}_7[x]/(f)$ and $\mathbb{Z}_7[x]/(g)$ isomorphic

Question

I have the following problem. Let $f(x) = x^3+5$ and $g(x) = x^3+2$. Prove that these two polynomials are irreducible over $\mathbb{Z}_7[x]$. Are the fields $\mathbb{Z}_7[x]/(f)$ and $\mathbb{Z}_7[x]/(g)$ isomorphic?

I have already prove by inspection that $f$ and $g$ doesn't have roots over $\mathbb{Z}_7[x]$, so they're irreducible. My questions are

• The fact that $\mathbb{Z}_7[x]/(f)$ and $\mathbb{Z}_7[x]/(g)$ are fields follow directly because $f$ and $g$ are irreducible so $(f)$ and $(g)$ are maximal?
• Is there an easier way to prove that the polynomials are irreducible?
• $\mathbb{Z}_7[x]/(f)$ and $\mathbb{Z}_7[x]/(g)$ are isomorphic because $(f), (g)$ generate the same ideal because they're monic?

Thanks

Answer 1

• Yes, $(f)$ and $(g)$ are maximal ideals, and as you were told their quotients are fields with the same number of elements.
  A direct approach to show that they are isomorphic is to consider the homomorphism which is identity on $\mathbb Z_7$ and maps $x$ to $-x$.

• I don't know an easier way to show that $f$ and $g$ are irreducible.

• No, $(f)\neq(g)$.