I don´t understand if there is a difference between the two concepts of
- Splitting Field of a Polynomial
- Extension Field Generated by roots of the Polynomial.
I know that given a polynomial $ p(x) \in \mathbb{F}[x]$ the Splitting Field of p is a extension field over $ \mathbb{F} $ in wich the polynomial p splits in linear factors.
On the other hand, I Know that if we have the roots of $ p(x) $, then we can form the extension field generated by the roots that includes $ \mathbb{F}$, so for example, if the roots of $ p(x) $ are $r_1, r_2 $ and $ r_3$ then we can form $ \mathbb{F}[r_1, r_2, r_3]$ wich is a extension field over $ \mathbb{F}$ generated by the roots of the polynomial.
By the way, is so clearly that the Splitting Field of $ p(x) $ is included (as subset) in $ \mathbb{F}[r_1, r_2, r_3]$ , but, is there any difference between the Splitting Field of $ p(x) $ and $ \mathbb{F}[r_1, r_2, r_3]$ ?
Is (the Splitting Field of a Polynomial $ p(x) \in \mathbb{F}[x]$) = (The extension Field over $ \mathbb{F}$ generated by the roots of $p(x) $) ?