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Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }.

I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($\pm1$). I don't know whether this solution is correct or accurate but I'm clueless for above problem.

Edit: The chapter was on matrices so I assumed the author was asking for such matrices, but author just said "Find all units" (He may have meant elements, idk).

prakasht
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    Welcome to stackexchange. Where are there matrices and determinants in these objects? Please [edit] the question to clarify, and show us what you did for $\mathbb{Z}$. – Ethan Bolker Jan 21 '19 at 15:05
  • Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p – prakasht Jan 21 '19 at 15:24

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You are right. With $$1=\begin{pmatrix}1&0\\0&1\end{pmatrix}, i=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$$ you can write any $a+bi\mid a,b\in\Bbb Z$ as $$\begin{pmatrix}a&-b\\b&a\end{pmatrix}$$

That is $\Bbb Z[i]\cong\left\{\begin{pmatrix}a&-b\\b&a\end{pmatrix}\mid a,b\in\Bbb Z\right\}$. Can you take it from here?

cansomeonehelpmeout
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