Units in ring theory

Question

How to determine $U( Z[i])$

I tried this

$$(a+bi)(c+di) = 1,$$

where $a,b,c,d$ are integers. and compared real no. With real

Did you even try to Google this? There are plenty of arguments showing how to determine the units of $\mathbb{Z}[i]$. For example, this: http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf — Sir_Math_Cat, Feb 21 '18 at 19:14
Also, this exact question had already been answered here: https://math.stackexchange.com/questions/108071/units-of-gaussian-integers — Sir_Math_Cat, Feb 21 '18 at 19:16

score 2 · Answer 1 · answered Feb 21 '18 at 19:13

2

Hint: note that if $a+bi$ is invertible, then the corresponding complex number must have norm $1$.

(In general, considering the norm is very often a good thing to do in this ring.)

answered Feb 21 '18 at 19:13

Patrick Stevens

score 1 · Answer 2 · answered Feb 21 '18 at 19:13

1

You can show that if $(a+bi)(c+di)$ is an integer, then $c=a, d=-a$. So the problem reduces to solving $$(a+bi)(a-bi)=a^2+b^2=1$$

What can you say about the integers $a,b$ if the above should be true?

answered Feb 21 '18 at 19:13

2 Answers2