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  1. What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers.
  2. Check whether $(1+i)$ is prime or maximal ideal.

Can someone help me please. I have no idea how to crack this problem. Thanks for your time.

gumti
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  • A useful tool to work with those numbers is to take their norm squared; $|a+bi|^2=a^2+b^2$. This allows you to pass information back and forth from factorization of those numbers to factorization of integers. Another tool is 'long division' which also holds there by leaving a remainder that has smaller norm than the divisor. – OR. Jul 15 '13 at 02:13
  • ok. <1+i> is a proper ideal.am I right?but how can I check it is prime or maximal – gumti Jul 15 '13 at 02:18
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    Because you have long division (people call this being an Euclidean domain) you can do the same proof as in the integers to show that all ideals are generated by one number. This helps for listing the ideals you want. – OR. Jul 15 '13 at 02:21
  • still could not get it. – gumti Jul 15 '13 at 02:33

1 Answers1

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$\mathbb Z[i]$ is a Euclidean domain, and thus a principal ideal domain. So every ideal is generated by a single element, this should help you give a description of the proper ideals. The prime ideals are related then to the prime elements which are known as Gaussian primes. Have a look at this for more information.

OR.
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Ittay Weiss
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