1

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$?

I am having a hard time understanding what the ring $\mathbb{R}/I$ should be. I know the formal definitions of quotient ring and the ring operations in it. However I cant make much of the structure of quotient rings. Could anyone make this simple for me?

user26857
  • 52,094
Miz
  • 2,739

2 Answers2

2

Hint: Instead of thinking about the quotient directly, first write it down in a familiar form using third isomorphism theorem. After that, use Chinese Remainder Theorem to conclude.

I'm travelling, so will be able to shed more light after a while.

Hmm.
  • 3,052
  • 1
    No need to use CRT since we don't need the ring structure to find the cardinality. – user26857 Jun 15 '16 at 09:43
  • @user26857 I wanted to write down the quotient explicitly. But of course as far as the OP concerned, no need to use CRT. – Hmm. Jun 15 '16 at 09:46
2

This is the same structure as $ \mathbb{Z}_2[x]/((x^2 + x + 1)(x^3 + x + 1)) $. By the Chinese remainder theorem, we have

$$ \mathbb{Z}_2[x]/((x^2 + x + 1)(x^3 + x + 1)) \cong \mathbb{Z}_2[x]/(x^2 + x + 1) \times \mathbb{Z}_2[x]/(x^3 + x + 1) = \mathbb{F}_4 \times \mathbb{F}_8 $$

The cardinality of this structure is clearly $ 32 $.

Ege Erdil
  • 17,747
  • No need to use CRT since we don't need the ring structure to find the cardinality. – user26857 Jun 15 '16 at 09:45
  • Sure, there is no harm in doing so, though :P – Ege Erdil Jun 15 '16 at 09:45
  • @Starfall Could you please explain why $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ has the same structure as $\mathbb{Z}_2[x]/((x^2+x+1)(x^3+x+1))$? – Miz Jun 13 '17 at 13:44
  • @Miz It doesn't, because you didn't mod out by $ 2 $ yet. – Ege Erdil Jun 13 '17 at 13:45
  • @Starfall I'm dont seem to understand what you mean by "This is the same structure as $ \mathbb{Z}_2[x]/((x^2 + x + 1)(x^3 + x + 1)) $" Could you please explain? – Miz Jun 13 '17 at 13:47
  • @Miz $ R/I $ is isomorphic to $ \mathbb Z_2[x]/((x^2+x+1)(x^3+x+1)) $. – Ege Erdil Jun 13 '17 at 13:48
  • @Starfall Would I be accurate in writing $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1)) = \mathbb{Z}[x]/(x^2+x+1) \times \mathbb{Z}[x]/(x^3+x+1)$? – Miz Jun 13 '17 at 13:55
  • @Miz No; those rings are not isomorphic. If you mod out by $ 3 $ on both sides, the ring on the left hand side has the nilpotent element $ (x^2 + x + 2)(x+2) $ of nilpotency index $ 3 $, whereas the ring on the right hand side has no elements with nilpotency index $ > 2 $. – Ege Erdil Jun 13 '17 at 14:05