Can anyone explain this? It makes sense when I look at it...
If f(x) has roots $ r_0, r_1, ... , r_n $ then f(x+a) has roots $ r_0 - a, r_1 - a, ... , r_n - a $.
So, the splitting field of f(x) over F must contain $ r_0, r_1, ... , r_n $.
And the splitting field of f(x+a) over F must contain $ r_0 - a, r_1 - a, ... , r_n - a $. But since F already contains a, this is the same as saying that the splitting field of f(x+a) over F must contain $ r_0, r_1, ... , r_n $.
So, the splitting field of f(x) over F is F adjoined any $ r_i $ not already in F. Same goes for f(x+a).
