Irreducibility
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a. Show that 1 + i is not a unit in Z[i]
b. Show that 2 is not irreducible in Z[i]
c. Show that 3 is irreducible in Z[i]
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Solution Summary
This solution shows how to solve a series of problem regarding irreducibility.
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Proof:
(a) Suppose 1 + i is a unit in Z[i], then we can find some a + bi in Z[i], such that
(1 + i)(a + bi) = (a - b) + (a + b)i = 1
Then we get
a - b = 1 and a + b = 0
We solve the system of equations and find that a = 1/2, b = -1/2
So a + bi = (1/2) - (1/2)i is not in Z[i], we get a contradiction.
Therefore, 1 + i is not a unit ...
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