6320-2022-08

Let \(G\) be a group, and \(N\) the subgroup generated by all elements of the form \(g^2\) for \(g \in G\). Prove that \(N\) is a normal subgroup containing the commutator subgroup of \(G\).

Prove that each 7-Sylow subgroup of the symmetric group \(S_{15}\) is abelian.

Let \(K\) be the splitting field of \(x^4-2x^2+4\) over \(\mathbb {Q}\). Determine \(\operatorname {Gal}(K/\mathbb {Q})\).

Determine the splitting field and the Galois group of \(x^9+1\) over \(\mathbb {F}_2\).

Let \(f(x)\) be an irreducible polynomial in \(\mathbb {Q}[x]\), with splitting field \(K\). Suppose \(\operatorname {Gal}(K/\mathbb {Q})\) is abelian. Prove or disprove: \(K = \mathbb {Q}(\alpha )\) for each root \(\alpha \) of \(f(x)\) in \(K\).