I have a problem which is to compute a simplified form for the eighth root of a polynomial of the form $x^2-ax+1$. In the solution here, it says that:
Notice that if $x$ satisfies the quadratic $x^2-ax+1=0$, then we have $$\begin{split} 0&=(x^2-ax+1)(x^2+ax+1)\\ &=x^4-(a^2-2)x^2+1. \end{split}$$ Clearly, then, the positive square roots of the quadratic $x^2-bx+1$ satisfy the quadratic $x^2-(b^2+2)^{1/2}x+1=0$....
What even are the "positive square roots" of a quadratic? Is this simply $\sqrt{x^2-bx+1}$? And how does the assertion in the last line follow?
The author presumably intended to write "the positive square roots of the roots of the quadratic $x^2-bx+1$ satisfy ...", a minor omission.
As far as the assertion, if $x$ satisfies $x^2-ax+1=0$, then $x^2$ satisfies $y^2-by+1=0$, where $b=a^2-2$.
Thus, squaring the roots leads to an equation of the same form (with $a$ replaced by $b=a^2-2$).
Reversing the process allows for "square-rooting" the roots.
Thus, the positive square roots of the roots of $y^2-by+1=0$ satisfy $x^2-ax+1=0$, where $a^2-2=b$, or equivalently (in the context where $a,b$ are positive), $a=\sqrt{b+2}$.
Note that the author wrote it incorrectly as $a=\sqrt{b^2+2}$, but it should be $a=\sqrt{b+2}$.