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Describe the ring $R = \mathbb Z_4[x]/((x^2+1)\mathbb Z_4[x])$ by

  1. listing all the cosets (for example by using coset representatives)
  2. describing the relations that hold between the elements in this ring, that is, describe the relations that hold between these cosets.
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    "Rumbled" is the word which springs to mind. It looks like someone is trying to get help with their homework without obvious effort. At least say how you've tried to solve it and if you have gotten anywhere.2012-04-04

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You need to use the division algorithm here; you can use it since $x^2 + 1$ is monic in the ring $\mathbb{Z}_4[x]$. For every polynomial $f \in \mathbb{Z}_4[x]$, by the division algorithm we can write it as

$f = (x^2 + 1)q(x) + r(x)$

where the degree of $r$ is bigger than or equal to zero, less than 2. You can now see that the cosets in the quotient are of the form

$(\text{linear polynomial}) + I$

where $I$ is the ideal generated by $x^2 + 1$. Now the linear polynomial can be written as $ax + b$ for $a,b \in \Bbb{Z}_4$. But then recall that $x^2 + 1 = 0$ in the quotient, so that we get ring a new ring (the quotient ring) where multiplication between cosets $A + I$ and $B + I$ is defined by $(A + I)(B+ I)= (AB) + I$ and where we have the relation $x^2 + 1 = 0$.

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    @MTurgeon I was totally mistaken, I have removed that from my answer now.2012-03-30