Here all fields are subfields of the complex numbers 0.1. Let F/K be a finite Galois extension and L/K a finite extension. Let FL be the smallest subfield of C containing both F and L. Show that FL/L is Galois, and there is a natural injective homomorphism Gal(FL/ L) ? Gal(F/ K) Now we view the first group as a subgroup of the second. Identify Gal (FL/L) as Gal(F/K') for some subfield K CKCF i.e. say exactly what K is, and of course prove your statement) Now assume L/K is also Galois. Show that Gal(FL/L) C Gal(F/K) is a normal subgroup, and identify the quotient group as a Galois group of some field extension. Show this quotient group is also a quotient of Gal(L/K), i.e. show there is a natural surjective homorphism Gal(L/ K) ? Gal(F/ K) / Gal(FL/ L)
