Es seien a , b {\displaystyle {}a,b} verschiedene quadratfreie Zahlen ≠ 1 {\displaystyle {}\neq 1} , die beide den Rest 1 {\displaystyle {}1} modulo 4 {\displaystyle {}4} haben. Es sei x = 1 + a 2 {\displaystyle {}x={\frac {1+{\sqrt {a}}}{2}}} und y = 1 + b 2 {\displaystyle {}y={\frac {1+{\sqrt {b}}}{2}}} . Zeige, dass z = 1 + a b 2 {\displaystyle {}z={\frac {1+{\sqrt {ab}}}{2}}} zu Z [ x , y ] {\displaystyle {}\mathbb {Z} [x,y]} gehört.
![{\displaystyle {}\mathbb {Z} [x,y]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/064d9b8e084fcd4d8c22cf34f29b910b32d2694d)