The Roots of the Polynomial Are the Elements of the Group

In a quadratic case if you look at the x squared plus one equals zero You know that that has these two roots that are i and minus i So if I Swapped each root with its negative. I sort of flipped the whole picture over But the polynomials can't tell it's still just x squaredplusone And then certain problems that we want to think about in terms of like well What if I had this root and I put it with rational numbers and I get a field? How do we write down elements there? We would say act on the roots of the group In a way that fixes the polynomial but permutes the roots Interesting so then we classify different