Examples of Euclid's Algorithm in Quadratic Integers

examplequadratic-integers

Example of Euclid’s Algorithm in $\mathbb{Z}[i]$

$\alpha =4+5i$ : $N(\alpha )=16+25=41$ $\beta =4-5i$ : $N(\beta )=16+25=41$

Thus, $\alpha ,\beta$ are primes in $\mathbb{Z}[i]$.

$$\begin{array}{rl}4+5i& =(4-5i)(i)+(-1+i)\\ 4-5i& =(-1+i)(-5)-1\\ -1+i& =(-1)(1-i)+0\end{array}$$

$GCD=-1$ (a unit!)

Use Euclid’s algorithm to write $(4+5i)x+(4-5i)y=1$ in $\mathbb{Z}[i]$:

$$\begin{array}{rl}-1& =(4-5i)-(-1+i)(-5),\\ & =(4-5i)+(-1+i)5,\\ & =(4-5i)(4+5i-(4-5i)i)5,\\ & =(4-5i)+(4+5i)5+(4-5i)(-5i),\\ & =(4+5i)5+(4-5i)(1-5i),\\ 1& =(4+5i)(-5)+(4-5i)(-1+5i).\end{array}$$ $$x=-5,y=(-1+5i)$$