Show that associates have the same norm, but that two Gaussian integers having same norm need not be associates.
Suppose, x & y are associates.
Then, x = u•y for some unit u of the integral domain.
So, N(x) = N(u•y) = N(u)N(y)
Since, u is a unit, so, N(u) = 1
So, N(x) = N(y)
Hence, associates have the same norm.
Let, x = 4+3i & y = 3+4i belong to Z[i]
Then, N(x) = 4²+3² = 25
&, N(y) = 3²+4² = 25
So, x & y have same norm.
But, they are not associates
Suopy, if they were associates, then,
(4+3i) = (a+bi)(3+4i) where, (a+bi) is a unit in Z[i]
Then, 3a - 4b = 4 & 4a + 3b = 3
Solving, a & b we get,
3(3a-4b) + 4(4a+3b) = 12 + 12
So, 25a = 24
So, a = 24/25 which is not in Z
Similarly, b is not in Z
So, a+bi does not belong to Z[i]
Hence, two Gaussian integers having same norm need not be associates.
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