Division in $\mathbb{Z}[i]$

Question

I am studying the ring theory and I got some point like $(2+3i)$ divides $(-1+5i)$ in the Eucledian domain $\mathbb{Z}[i]$. I do not know that how it can be possible. Please let me know about this.

Here we can reduce divisiion by complex integers to division by integers by rationalizing the denominator of the quotient. This is a special of the general method of simpler multiples. — Bill Dubuque, Oct 23 '19 at 14:10

score 6 · Accepted Answer · answered Oct 23 '19 at 09:54

6

Just do the division in $\mathbb{C}$: $$ \frac{-1+5i}{2+3i}=\frac{(-1+5i)(2-3i)}{4+9}=\frac{-2+10i+3i-15i^2}{13}=1+i $$ The quotient is in $\mathbb{Z}[i]$.

By the way, both $1+i$ and $2+3i$ are primes in $\mathbb{Z}[i]$.

answered Oct 23 '19 at 09:54

egreg

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Division in $\mathbb{Z}[i]$

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