It might be too little to care, but there is a term that isn't really defined in a math text, and I was looking if someone could provide a more complete understanding of it. For context:
$\Bbb{Q}(a_0, ... , a_{n-1})$ is the field of the rationals adjoined the coefficients $a_0, ..., a_{n-1}$ of a polynomial.
$x_1, ... , x_n$ are the roots of this polynomial.
The text: "The goal of solution by radicals is to extend $ℚ(a_0, ... , a_{n-1})$ by adjoining radicals until a field containing the roots $x_1, ... , x_n$"
The author gives the example of the quadratic equation and how one can have $x_1, x_2$ in the field extension of $\Bbb{Q}(a_0, ... , a_{n-1})$ the polynomial by adjoining the root of $a_1^2-4a_0$.
My question, I think, is rather simple. What exactly do they mean by "radical"? As in, radical of what? Do they mean only radicals of the coefficients? Or are we also talking about radicals of the roots? Do these have to be radicals of only rational numbers?
I would truly appreciate any help/thoughts!
It's radicals as in $\sqrt x$, $\sqrt[3]x$, and so on. And radicals of what? Well, radicals of the coefficients (not the roots) and more generally of the polynomial expressions obtained from the coefficients. And you can apply the radicals again to these new numbers.
So, in the case of quadratic equations $x^2+ax+b=0$, first you compute $a^2-b$. Then you compute $\Delta$, which is any square root of that number. Then the roots will be $\dfrac{\pm\Delta-a}2$.