I was reading some posts regarding the quotient Gaussian integers, and am now confused about how a quotient Gaussian integers can be represented/explained in the way of a polynomial ring (or a quotient of the ring of polynomials).

For example, I know $\mathbb{Z}[i]/\langle 1+3i\rangle\cong\mathbb{Z}_{10}$, but what is the isomorphism between $\mathbb{Z}[i]/\langle 1+3i\rangle$ and a quotient ring of polynomial, and why?

I appreciate if anyone could explain a bit. Thanks

  • $\begingroup$ This is the main idea here. Suppose $K\subseteq L$ are fields and $\theta\in L$ is algebraic over $K$. Let $m(x)\in K[x]$ be the minimal polynomial of $\theta$. Then, $K[\theta]\cong K[x]/(m(x))$. $\endgroup$ – tc1729 Aug 29 '14 at 0:59
  • $\begingroup$ Thanks Prasad, I am still confused about your explanation. Could you please give an example or introduce me a tutorial/book to read. Many thanks $\endgroup$ – Yi W Aug 29 '14 at 13:40

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