Algoritmo de Euclides
Exemplo 3
\(z=3+5\sqrt{2}\;\;\;\;\;\;\;\;\;\; w=2-\sqrt{2}\)
\[\begin{array}{l}
\;\;\;3+5\sqrt{2}=\left(2-\sqrt{2}\right)\left(q_{1}+q_{2}\sqrt{2}\right)+r_{1}+r_{2}\sqrt{2}\Leftrightarrow\\
\Leftrightarrow\frac{r_{1}+r_{2}\sqrt{2}}{2-\sqrt{2}}=\frac{3+5\sqrt{2}}{2-\sqrt{2}}-\left(q_{1}+q_{2}\sqrt{2}\right)\Leftrightarrow\\
\Leftrightarrow\frac{r_{1}+r_{2}\sqrt{2}}{2-\sqrt{2}}=\frac{\left(3+5\sqrt{2}\right)\left(2+\sqrt{2}\right)}{\left(2-\sqrt{2}\right)\left(2+\sqrt{2}\right)}-\left(q_{1}+q_{2}\sqrt{2}\right)\Leftrightarrow\\
\Leftrightarrow\frac{r_{1}+r_{2}\sqrt{2}}{2-\sqrt{2}}=8-q_{1}+\left(\frac{13}{2}-q_{2}\right)\sqrt{2}
\end{array}\]
Basta escolher \(q_{1}=8\)
e \(q_{2}=6\)
;
vem então
\(r_{1}+r_{2}\sqrt{2}=3+5\sqrt{2}-\left(2-\sqrt{2}\right)\left(8+6\sqrt{2}\right)=-1+\sqrt{2}\)
e temos
\(3+5\sqrt{2}=\left(2-\sqrt{2}\right)\left(8+6\sqrt{2}\right)+\left(-1+\sqrt{2}\right)\),
com
\(v\left(-1+\sqrt{2}\right)=1<2=v\left(2-\sqrt{2}\right)\)