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Define a function $\delta : \mathbb{Z}[i] \rightarrow \mathbb{N}$ by $\delta(a+bi)=a^2 +b^2$. Show that $\delta$ is a Euclidean valuation on the Gaussian Integers $\mathbb{Z}[i]$ i.e. For Gaussian Integers $a,b$ there exists Gaussian Integers $q,r$ such that $a=qb+r$ with either $r=0$ or $\delta(r)<\delta(b)$.

What I have so far: Let $a=a_{1}+a_{2}i$ and $b=b_{1}+b_{2}i$ be Gaussian Integers. Then $\frac{a}{b}=\frac{(a_{1}b_{1}+a_{2}b_{2})+(a_{2}b_{1}-a_{1}b_{2})i}{\delta(b)}$. Let $q$ be the Gaussian Integer which is closest to $\frac{a}{b}$ in the complex plane and $r=a-qb$. I then think I need to show that $r$ is suitably small, but I'm unsure how to do this. Any help would be greatly appreciated.

jackwo
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1 Answers1

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HINT: Let $\frac ab = r + si$ for some rational numbers $r,s$. Now let $q_1,q_2$ be the closest integers to $r,s$, respectively. Define $q=q_1 + q_2i$ and $r = a - bq$. Using the fact that $\mid \;r - q_1 \mid \le \frac 12$ and $\mid \; s - q_2 \mid \le \frac 12$, prove that unless $r=0$, then $\delta(r) < \delta(b)$.

Stefan4024
  • 35,843