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Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}....

Definition:In the complex numbers, let J denote the set, {x+y√3i :x and y are in Z}. J is an integral domain containing Z. If a is in J, then N(a) is a non-negative member of Z. If a
and b are in J and a|b in J, then N(a)|N(b) in Z. The units of J are 1, -1

Question:If a and b are in J and ab = 2, then prove one of a and b is a unit. Thus, 2 is prime in J. Show −2 is also prime.

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Answer #1

Then

As if neither of a and b are units we must have

If as is not possible which makes which is also impossible

So meaning at one of them is a unit

Similarly, if and similar logic tells us that

meaning at one of them is a unit

Thus, 2 and -2 are primes in J

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