Defining equation for golden ratio; quadratic fields
Definition: If is an irrational number in , then the equation is called the defining equation for if satisfies the equation and a, b, and c are integers, , and .
Definition: If is an irrational number in , then the equation is called the defining equation for if satisfies the equation and a, b, and c are integers, , and .
Find a defining equation for the golden ratio .
How do you approach this problem?
The solutions to such an equation are :
Since a and b are integers, the only way to get this is .
So you can see that will also be a root. (The reasoning may be similar to the ones involving complex roots).