gaussian integers quotient ring

Asked Apr 18 '16 at 16:41

Active Oct 22 '21 at 13:46

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Let $R$ denote the factor ring $\mathbb{Z}[i]/(1+3i)$. Show that $i-3 \in (1+3i)$ and that $i+(1+3i) = 3 + (1+ 3i)$ in R. I am unsure how to find the elements of this factor ring?

I know how to find the elements of, for example $\mathbb{F}_{3}[x]/(x^{2}+1)$, but unsure about this example with Gaussian integers.

asked Apr 18 '16 at 16:41

DMSTA

1

So in the factor ring, $1 + 3i = 0$, or $1 = -3i$, or $i = 3$. – M. Vinay Apr 18 '16 at 16:44
Note that $i-3=i(1+3i)$ and hence, by definition, your first statement. This implies that $[i-3]=[0]$ in $R$ and hence that $[i]=[3]$, which is your second statement. – User3773 Apr 18 '16 at 16:45
$(1+3i)$ and $(3-i)$ generate the same ideal, so this is an exact duplicate. – Jyrki Lahtonen Oct 23 '21 at 02:45

gaussian integers quotient ring

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