6320-2022-05

Show there is no simple group of order \(2304 = 2^8 \cdot 3^2 = 256 \cdot 9\).

Let \(K = \mathbb {Q}[\sqrt {2},\sqrt {3},\sqrt {5}]\). Show that \(K\) is Galois over \(\mathbb {Q}\) and identify \(\operatorname {Gal}(K/\mathbb {Q})\) up to isomorphism.

Classify the groups of order 28 up to isomorphism.

Let \(F\) be a field of characteristic zero and let \(E/F\) be a finite Galois extension. Suppose that \(E = F[\alpha ]\). Suppose that there exists \(\sigma \in G = \operatorname {Gal}(E/F)\) with \(\sigma (\alpha )=-\alpha \). Prove that \(E \ne F[\alpha ^2]\).

Let \(L\) be the splitting field of \((x^3+2x+1)(x^3+x^2+2)(x^2+1)\) over \(\mathbb {F}_3\). How many proper subfields does \(L\) have?