Units of the Gaussian integers
Moderate Ring Theory question for IIT JAM · CSIR-NET. Choose an option, check your answer, then read the detailed solution.
Question RT-011
Let $\mathbb Z[i]=\{a+bi:a,b\in\mathbb Z\}$ be the ring of Gaussian integers, with the norm $N(a+bi)=a^2+b^2$. Recall that $u\in\mathbb Z[i]$ is a unit if and only if $N(u)=1$. Which of the following correctly describes the units of $\mathbb Z[i]$?
Detailed Solution
A unit $u=a+bi$ must satisfy $N(u)=a^2+b^2=1$ with $a,b\in\mathbb Z$. The only integer solutions are $(a,b)\in\{(1,0),(-1,0),(0,1),(0,-1)\}$, giving exactly the four units $\{1,-1,i,-i\}$ — this proves (A). $\mathbb Z[i]$ is not a field: for instance $N(2)=4\neq1$, so $2$ has no inverse in $\mathbb Z[i]$, ruling out (B). Option (C) overcounts: it would include $1+i$, but $N(1+i)=1+1=2\neq1$, so $1+i$ is not a unit. Option (D) is false: $N(2+i)=4+1=5\neq1$, so $2+i$ is not a unit. Thus the correct option is $\boxed{\mathrm{A}}$.Concepts tested: Gaussian integersUnitsNorm
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