1. ## Defining equation for golden ratio; quadratic fields

Definition: If $\alpha$ is an irrational number in $\mathbf{Q}\left(\sqrt{d}\right)$, then the equation $ax^{2}+bx+c=0$ is called the defining equation for $\alpha$ if $\alpha$ satisfies the equation and a, b, and c are integers, $(a,b,c)=1$, and $a>0$.

Find a defining equation for the golden ratio $\dfrac{1+\sqrt{5}}{2}$.

How do you approach this problem?

2. Hello,
Originally Posted by Pn0yS0ld13r
Definition: If $\alpha$ is an irrational number in $\mathbf{Q}\left(\sqrt{d}\right)$, then the equation $ax^{2}+bx+c=0$ is called the defining equation for $\alpha$ if $\alpha$ satisfies the equation and a, b, and c are integers, $(a,b,c)=1$, and $a>0$.

Find a defining equation for the golden ratio $\dfrac{1+\sqrt{5}}{2}$.

How do you approach this problem?
The solutions to such an equation are :

$x=\frac{-b {\color{red}\pm} \sqrt{b^2-4ac}}{2a}$

Since a and b are integers, the only way to get this $\sqrt{5}$ is $\sqrt{b^2-4ac}$.
So you can see that $\frac{1{\color{red}-}\sqrt{5}}{2}$ will also be a root. (The reasoning may be similar to the ones involving complex roots).

Develop $\left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)$ and identify a,b and c

3. Thank you Moo.

I can't believe I didn't get this before...