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I'm a bit confused if I'm going about this problem correctly, trying to find the cosets of G in $Z_{3}[x]$ where G := {$(x^{3}+ x^{2} + 2x + 1)(g(x))$: with $g(x)$ in $Z_{3}[x]$}

I've been given that $1 + G$ is a coset since the resulting element is still within G. What I don't understand is how to find the rest of the cosets.

For example would $x^{3}+ x^{2} + G$ be another coset. If i'm understanding this correctly then I believe I would have to write out 14 cosets which seems like a lot.

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$G$ is the principal ideal generated by $\phi(x)=x^3+x^2+2x+1$ over $\Bbb Z_3$. Each coset $u(x)+G$ has the form $ax^2+bx+c+G$ where $a$, $b$, $c\in\Bbb Z_3$. So there are $3^3=27$ cosets; I don't feel the need to write out all of them, but perhaps you need to.

To see this note that by the division algorithm, if $f$ is a polynomial, then $f(x)=q(x)\phi(x)+r(x)$ where $r(x)$ has degree $<3$. Then $f(x)+G=r(x)+G$.

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