ABSTRACT
In Chapter 7, we looked at some examples of symmetry groups of equations. A symmetry of an equation was defined as a permutation of its roots that preserved any algebraic relations among them. We were not very exact about what we meant by “algebraic relations among the roots.” In the last chapter, we made this more precise and saw that the structure of the splitting field of an equation reflects these algebraic relations. So a natural way to define what a symmetry of an equation should be is to say that it should be a mapping of the splitting field to itself that preserves the structure of the field and fixes the coefficients of the equation. As we shall show shortly, this implies that it will permute the roots. To put it more succinctly: if we have a polynomial f with coefficients in a field F, with a splitting field E/F, then a symmetry is an automorphism of E that fixes F.
