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 by $x\mapsto x-\frac1x$ sends $\phi_n$ to $\phi_{n-1}$ for each $n\ge1$ and so there is the nested chain of algebraic number fields $$\mathbb Q=\mathbb Q[\phi_1]\subseteq\mathbb Q[\phi_2]\subseteq\mathbb Q[\phi_3]\subseteq\dotsb.$$ 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?

Added following R.P.'s comment: I don't ask for a solution to a homework problem. More useful would be a recommendation of a text book or article where this kind of problem is addressed. There are some results with similar flavor in Kaplansky's book Fields and Rings (Chicago, 2nd ed. 1972) which look superficially promising such as Theorems 53 and 54 of the Miscellany that is section 12 of Part I. Some additional references would be a huge help and enough for me to consider the question closed.