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Math Help - gaussian integers/integral domains

  1. #1
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    gaussian integers/integral domains

    I am having some trouble with these problems:

    1. Find all solutions of (x^2) - x + 2 = 0 over Z_3[i], where Z_3[i] =
    {a + bi, where a, b are an element of Z_3} = {0, 1, 2, i, 1 + i, 2 + i, 2i, 1 + 2i, 2 + 2i}, where i^2 = -1 (the ring of Gaussian integers modulo 3).

    2. Find all units and zero divisors of Z_3 "direct sum" Z_6.

    3. Let F be a finite field with n elements. Prove that x^(n-1) = 1 for all nonzero x in F.

    4. Is Z_2[i] a field? Is it an integral domain? Note that Z_2[i] = {a + bi, where a, b are an element of Z_2} = {0, 1, i, 1 + i}, where i^2 = -1 (the ring of gaussian integers modulo 2).

    For #4, I'm fairly certain it's not an integral domain since (1 + i)(1 + i) = 2i = 0 in mod 2, making (1+i) a zero divisor. I'm not sure how to show if it is a field or not though.

    Thanks guys.
    Last edited by eigenvector11; November 27th 2007 at 05:44 PM.
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  2. #2
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    Quote Originally Posted by eigenvector11 View Post
    1. Find all solutions of (x^2) - x + 2 = 0 over Z_3[i], where Z_3[i] =
    {a + bi, where a, b are an element of Z_3} = {0, 1, 2, i, 1 + i, 2 + i, 2i, 1 + 2i, 2 + 2i}, where i^2 = -1 (the ring of Gaussian integers modulo 3).
    Any element can be written as a+bi where a,b are the equivalence classes mod 3. There are 9 such possibilities just check each one.
    2. Find all units and zero divisors of Z_3 "direct sum" Z_6.
    A unit (a,b) in this ring is such that (a,b)(x,y)=(1,1)\implies (ax,by)=(1,1). Thus the pair is invertible if and only if a \mbox{ and } b are invertible. Use this to complete the problem.

    For zero divisors note that if (a,b)(c,d)=(0,0)\implies (ac,bd)=(0,0) but a,c\in \mathbb{Z}_3 are in a field and they are no zero divisors hence a,c=0. But bd=0 are in \mathbb{Z}_6 so b=2,d=3 \mbox{ and }b=3,d=2 are zero divisors.

    3. Let F be a finite field with n elements. Prove that x^(n-1) = 1 for all nonzero x in F.
    The multiplicative set of F of non-zero elements forms a group under multiplication. Since it has n-1 elements it means x^(n-1) = 1 (property of groups).

    4. Is Z_2[i] a field? Is it an integral domain? Note that Z_2[i] = {a + bi, where a, b are an element of Z_2} = {0, 1, i, 1 + i}, where i^2 = -1 (the ring of gaussian integers modulo 2).
    Just check the defitions.
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