What are the unit elements in $\Bbb{Z}[i]$?

Question

What are the unit elements in $\Bbb{Z}[i]$, where $\Bbb{Z}[i]$ is defined to be the set of Gaussian integers ? My progress: My mentor gave this problem and he said to use determinants and scalefactor. Then I was able to proceed and got the unit elements in $\Bbb{Z}[i]=1,-1,i,-i$ , which is correct . However, I wonder if there is some other way to proceed .

Any solution or hint is appreciated .

Thanks in advance.

The usual approach is to consider $|z|^2$ for $z\in\Bbb Z[i]$. — Angina Seng, Jul 28 '20 at 02:22
What is $|z|^2$ if $z=a+bi$? What is $|z|^2$ if $z$ is a unit? — J. W. Tanner, Jul 28 '20 at 02:26
we have N(z)=a^2+b^2, if z is a unit then it must divide 1 , so N(z)|+-1,is it okay till now ? — Sunaina Pati, Jul 28 '20 at 02:30
Okay I got it thanks :) . we have a^2+b^2= 1 or -1 . proceeding we get z=1,−1,,− — Sunaina Pati, Jul 28 '20 at 02:32
and $N(z)$ can't be $-1$ because $a^2+b^2$ can't be negative — J. W. Tanner, Jul 28 '20 at 02:39

score 4 · Accepted Answer · answered Jul 28 '20 at 02:40

If $a+bi$ is a unit ($a,b$ are integers) then for some integers $c,d$ we have $$(a+bi)(c+di)=1.$$ Then the norms of the LHS and RHS should be the same. To get the norm of the LHS, we multiply by $a-bi$ and $c-di$, we get $(a^2+b^2)(c^2+d^2)$. So $(a^2+b^2)(c^2+d^2)=1$. Since both factors are non-negative integers, both are equal to 1. So $|a+bi|=1$. Conversely if $|a+bi|=1$ then $a^2+b^2=(a+bi)(a-bi)=1$ so $a+bi$ is a unit. For integers $a,b$ $a^2+b^2=1$ is possible iff $a=\pm1, b=0$ or $a=0, b=\pm1$. Thus there are four units: $1,-1, i, -i$.

score 1 · Answer 2 · answered Jul 28 '20 at 03:37

1

Hint: If the norm of a gaussian integer is defined by $N(a+bi)=a^2+b^2$, then this norm is multiplicative. That is, $N(xy)=N(x)N(y)$.

answered Jul 28 '20 at 03:37

What are the unit elements in $\Bbb{Z}[i]$?

2 Answers2