Remark $1.3$ of the pdf here states that
There are three monic quadratic factors with $2^\frac{1}{4}$ as a root, but only one of them, $X^2−2^\frac{1}{2}$, has coefficients in $\mathbb{Q}\left (2^\frac{1}{4}\right )$.
I could not get why other factors cannot be minimal polynomial of $2^\frac{1}{4}$ and why should the coefficents must lie in $\mathbb{Q}\left (2^\frac{1}{4}\right )$. Is it because the extension is not Normal and since we want the coefficents to lie inside $F$, we are showing that if it doesn't lie in $\mathbb{Q}\left (2^\frac{1}{4}\right )$, hence it doesn't lie in $F$ as well?
The argument introduces the minimal polynomal $P$ of $\sqrt[4]{2}$ over $F\subset \mathbb{Q}[\sqrt[4]{2}]$. So this polynomial must have coefficients in $F$ (by definition of a minimal polynomial), and therefore in $\mathbb{Q}[\sqrt[4]{2}]$, since $F\subset \mathbb{Q}[\sqrt[4]{2}]$.
Any factor of $X^4-2$ in $F$ must in particular be a factor in $\mathbb{C}$, and you can easily check what are the quadratic factors of $X^2-4$ in $\mathbb{C}$, and see for yourself that the only one with coefficients in $\mathbb{Q}[\sqrt[4]{2}]$ (and therefore the only one with coefficients in $F$) is indeed $X^2-\sqrt{2}$.