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Question:

Let $R = \{a + bi | a,b \in \Bbb Z \}$, let $M = \{x(2 + i) | x \in R\}$. Prove M is a maximal ideal of R.

I just started learning about ideals so I apologize for asking a basic question, please describe steps as simply as possible.

After reading online and related questions on here, I was trying to first prove M is an ideal and then prove R/M is a field implying M is a maximal ideal. Is this misguided? And, if not, how would I go about doing step two?

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It's not misguided. You could prove that $R/M$ is a field by thinking about what field it would be. Hint: rewrite $R = \mathbb{Z}[i]$ as $\mathbb{Z}[T]/(T^2+1)$, which makes the quotient easier to calculate.

You could also take the easier approach of calculating the number of elements of $R/M$, which tells you quite a bit about its structure.

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  • $\begingroup$ How would I calculate the number of elements? $\endgroup$ Commented Oct 30, 2014 at 7:46
  • $\begingroup$ @User52525 In short, the Euclidean algorithm. How many remainders can there be on division by $2+i$? You could start out dividing by $(2+i)(2-i) = 5$, then refine your search. $\endgroup$ Commented Oct 30, 2014 at 7:57

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