The Fundamental Theorem of Arithmetic for $\mathbb Z[i]$

Question

I understand how to prove the Fundamental Theory of Arithmetic, but I do not understand how to further articulate it to the point where it applies for $\mathbb Z[I]$ (the Gaussian integers).

I started by first defining norm/modulus and proving basic properties.

Next, I am going to show that the division algorithm works in $\mathbb Z[I]$.

After this, I will then define gcd and show that its two definitions agree. This here will complete my proof.

My question is, where do I begin to show that the division algorithm works in $\mathbb Z[I]$?

You start proving the division algorithm by stating it in a form that makes sense in this context. You can't simply say that when dividing by $q > 0$ you can guarantee a remainder between $0$ and $q-1$. — Ethan Bolker, Dec 07 '20 at 23:06
By $\mathbb{Z}(i)$ do you mean the Gaussian integers, or just the integers? — Noah Solomon, Dec 07 '20 at 23:16
@PaigeJordan I would suggest you use the notation $\mathbb{Z}$ for the set of integers, as $\mathbb{Z}(i)$ usually denotes the gaussian integers. — Noah Solomon, Dec 07 '20 at 23:30
@NoahSolomon I apologize, it is referring to the Gaussian integers, I was reading my notes incorrectly. — Paige Jordan, Dec 08 '20 at 00:13
The Gaussian integers are usually denoted $\mathbb{Z}[i]$ (with brackets). Normally, the use of parentheses denotes a field of fractions of some kind. — Arturo Magidin, Dec 08 '20 at 00:57
The geometric argument used to show that $\mathbb{Z}[\zeta_3]$ is Euclidean/PID/UFD also words for $\mathbb{Z}[i]$. See here — Arturo Magidin, Dec 08 '20 at 00:59

score 1 · Answer 1 · answered Dec 08 '20 at 00:30

Say you want to divide $b$ into $a$. Let $q$ be the true quotient $a/b$ rounded off to the nearest Gaussian integer. Then $a/b-q$ has real and imaginary components with absolute value less than or equal to $1/2.$ From here, you should show that $|r|^2 <|b|^2,$ where $r=a-bq.$

The Fundamental Theorem of Arithmetic for $\mathbb Z[i]$

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