Question

(i) The total number of units in j[i], the ring of Gaussian integers, is equal to
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Answer 1

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Answer 2

The total number of units in j[i], the ring of Gaussian integers is:

Explanation:

• Units are those elements in a ring that are invertible. Assume a+ bi is a unit.
• Then ∃ c+ di ∈Z[i] such that (a+ bi)(c +di)=1. This implies
• {ac−bd=1bc+ad=0
• Now solve this system and remember that a, b, c, d∈ Z.
• We get from the second equation that bc=−ad. This means that, from the first equation,
• If b≠0, then, d(a2+b2)=−b.
• This leaves with two possibilities: d=−1 0r d=−b.
• But, if d=−b then, we have that c=a and a2+b2=1. Now since we have been given that b≠0 the only solution for a2+b2=1 with a, b∈ Z will be a=0 and b=±1.
• Now, if d=−1, then note that if c=0, we immediately have that b=1.
• The last equation we derived from the system gives you that, b(1−b)=a2 which has no integral solutions if b≠0.
• For the last case, if b=0, then, the Gaussian integer is just a and a has an inverse in Z[i] iff a=±1. This is because, the multiplicative inverse is 1a, which will be in Z iff a=±1.