Only the highlighted Portion, thank you.
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) 0.2. Let f KX be an rreducible polyn, whose Galois group is a non Abelian simple group (simple means it has no non-trivial normal subgroups). Use the results of problem (1) to prove that if L/K is a Galois extension with an Abelian Galois group, that f(x) E L[X] is irreducible, and has the same Galois group. Ie. Gal(f/L) = Gal(f/K). This is proven in Artin, Proposition 9.8 in the section on quintics. But his argument is overly complicated, if you repeat it, you will get some, but not full, credit Let g E K[X] be a quntic with Galois group either A5 or S5 and let L/K be an Abelian Galois extensions. Use (1) to show that the Galois group of g over L is either A5 or S5 (this includes the possibility that it started out S5 and then became A5). You may use the fact that A5 is simple, and that the only non-trivial proper normal subgroup of Ss is As. Prove that g(X) is irreducible over L. Now show if you have any tower of extensions with Li+1/L, an Abelian Galois extension, that g(X) E LIX] is irreducible