The higher Golden Ratios $(\phi_n)_{n=0,1,2,3,\dots}$ are the real numbers defined recursively by setting $\phi_0=0$ and for $n\ge1$, $\phi_n$ is the positive (real) root of $X^2-\phi_nX-1$. We find that $\phi_1=1$ and $\phi_2=\frac{1+\sqrt5}{2}$. In particular, $\phi_2$ is equal to the classical Golden Ratio. Moreover the sequence $(\phi_n)_{n=0,1,2,\dots}$ is strictly increasing. The function $\mathbb R^\times\buildrel f\over\to\mathbb R$ defined $x\mapsto x-\frac1x$ sends $\phi_n$ to $\phi_{n-1}$ for each $n\ge1$ and so there is the nested chain of algebraic number fields $$\mathbb Q=\mathbb Q[\phi_1]\subseteq\mathbb Q[\phi_2]\subseteq\mathbb Q[\phi_3]\subseteq\dots.$$ For each $n\ge1$ we have $$|\mathbb Q[\phi_{n+1}]:\mathbb Q[\phi_n]|= 1\textrm{ or }2.$$ I guess it is always $2$. How to prove this guess??