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Math Help - Quotient Ring of the Gaussian Integers

  1. #1
    MHF Contributor chiph588@'s Avatar
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    Quotient Ring of the Gaussian Integers

    Given  p \equiv 3 \mod{4} is prime in  \mathbb{Z} , show  \mathbb{Z}[i]/(p) has order  p^2 .
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  2. #2
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    Quote Originally Posted by chiph588@ View Post
    Given  p \equiv 3 \mod{4} is prime in  \mathbb{Z} , show  \mathbb{Z}[i]/(p) has order  p^2 .
    An odd prime p has a form either 1(mod 4) or 3(mod 4). We can see that 1(mod 4) is reducible in Z[i] (For example, 5=(1+2i)(1-2i)). An odd prime p is irreducible iff it has the form 3(mod 4).

    We know that Z[i] is an integral domain and a PID. We also know that  p \equiv 3 (mod{4}) is irreducible. Thus <p> is maximal ideal in Z[i]. It follows that Z[i]/<p> is a field having p^2 elements.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by aliceinwonderland View Post
    It follows that Z[i]/<p> is a field having p^2 elements.
    I don't quite see how that follows.
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  4. #4
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    Quote Originally Posted by chiph588@ View Post
    I don't quite see how that follows.
    An element of Z[i]/<p> has the form a+bi. The number of choices of a is p and the number of choices for b is p. So it has p^2 elements.

    For example, let p =3. Then, Z[i]/<3>= {0, 1,2, i, 2i, 1+i, 1+2i, 2+i, 2+2i}. It has 9 elements. You can check other cases and verify them.
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