# Describe the ring and the cosets.

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.
• I see no question. – Najib Idrissi Mar 30 '12 at 12:51
• Can't you just ask Dr Smoktunowicz if you are stuck? – user28109 Apr 1 '12 at 22:55
• @AFellowStudent If only he could pronounce her name. – Alex Youcis Apr 1 '12 at 23:04
• @AFellowStudent: Who is Dr Smoktunowicz? – spohreis Apr 1 '12 at 23:12

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 $x^2 + 1 = 0$.
• "evaluating the left and right at $x=i$"? The complex number $i$? Anyway, if you are looking for a square root of $-1$ in $\mathbb{Z}/4\mathbb{Z}$, it doesn't exist. – M Turgeon Mar 30 '12 at 12:59