Question

Show that associates have the same norm, but that two Gaussian integers having same norm need not be associates.

Answer 1

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.