Now, if we have a field F which is a field extension of Q then we have a collection G of Q-automorphisms of F. This collection G is a group (with the operation defined by: if f and g are in G, i.e. they are Q-automorphisms of F, then f⋅g is a Q-automorphism defined by (f⋅g)(x)=f(g(x)) - check that this really is a group). It is called the Galois group of the field extension F over Q , usually written Gal(F/Q). If F is the splitting field of a polynomial p(x) then G is called the Galois group of the polynomial p(x), usually written Gal(p).
