Chapter 33

Explain why every abelian group is, trivially, a solvable group.

Find radical extensions of \(\mathbb{Q}\) containing the following complex numbers: (a) \((\sqrt{5}-\sqrt[5]{2}) /(\sqrt[4]{3}+\sqrt[3]{4})\) (b) \(\sqrt{(1-\sqrt[4]{2)} / \sqrt{1-\sqrt{5}}}\) (c) \(\sqrt[3]{(\sqrt{3}-2 i)^{3} /(i-\sqrt{11})}\)

Let \(G\) be a solvable group, with a solvable series \(H_{0}, \ldots, H_{n} .\) Let \(K\) be a sul oup of \(G\). Show that \(J_{0}=K \cap H_{0}, \ldots, J_{n}=K \cap H_{n}\) is a normal series of \(K\).

Show that the following polynomials in \(\mathbb{Q}[x]\) are not solvable by radicals: (a) \(2 x^{5}-5 x^{4}+5\) (b) \(x^{5}-4 x^{2}+2\) (c) \(x^{5}-4 x^{4}+2 x+2\)

Show that \(a x^{8}+b x^{6}+c x^{4}+d x^{2}+e\) is solvable by radicals over any field.(HINT: Let \(y=x^{2}\); use the fact that every fourth-degree polynomial is solvable by radicals.)

If \(H \triangleleft K \triangleleft G\), then \(G / K\) is a homomorphic image of \(G / H\).

Conclude: If \(x^{p}-a\) is not irreducible in \(F[x]\), it has a root (namely \(\left.b^{s} a^{\prime}\right)\) in \(F\). We have proved: \(x^{p}-\) a either has a root in \(F\) or is irreducible over \(F\).