7 Problem Sheet 7:
Cyclic extensions
Exercise 7.1 Use Kummer Theory to show the following polynomials are irreducible over the given fields:
\(\;(a)\;\) \(x^4+2\in\mathbb{F}_{13}[x]\),
\(\;(b)\;\) \(x^7+\theta\in\mathbb{F}_2(\theta)[x]\) where \(\theta\) is a root of \(x^3+x+1\in\mathbb{F}_2[x]\),
\(\;(c)\;\) \(x^4-(1+\theta)\in\mathbb{F}_9[x]\) where \(\theta\in\mathbb{F}_9\) satisfies \(\theta^2+1=0\).
Exercise 7.2 Let \(L=\mathbb{Q}(\zeta_{13})\) where \(\zeta_{13}=e^{2\pi i/13}\).
\(\;(a)\;\) Prove that \(L\) contains a unique subfield \(K_6\) with \([K_6:\mathbb{Q}]=6\) and that \(L/K_6\) is a cyclic extension. Explain why we can apply Kummer Theory to deduce \(L=K_6(\sqrt{a})\) for some \(a\in K_6\).
\(\;(b)\;\) Prove that \(L\) contains a unique subfield \(K_4\) with \([K_4:\mathbb{Q}]=4\) and \(L/K_4\) is a cyclic extension. Explain why we cannot apply Kummer Theory to deduce \(L=K_4(\sqrt[3]{b})\) for some \(b\in K_4\). Furthermore, prove that no such \(b\) exists.
Exercise 7.3 Let \(L=\mathbb{Q}(\zeta_{20})\) where \(\zeta_{20}=e^{2\pi i/20}\) and \(K=\mathbb{Q}(i)\).
\(\;(a)\;\) Show that \(L=K(\zeta_5)\) where \(\zeta_5=e^{2\pi i/5}\) and prove that \(\Gal(L/K)\cong\mathbb{Z}/4\).
\(\;(b)\;\) Explain why there exists \(a\in K\) such that \(L=K(\sqrt[4]{a})\) and use the Lagrange resolvent construction to show that \(a=5(1-2i)^2=-15-20i\) is a suitable choice.
\(\;(c)\;\) Prove that \(\mathbb{Q}(\sqrt{-5})\subset L\) and find \(\Gal(L/\mathbb{Q}(\sqrt{-5}))\).
Exercise 7.4 Let \(L=\mathbb{Q}(\zeta)\) where \(\zeta=\zeta_{25}=e^{2\pi i/25}\).
\(\;(a)\;\) Prove that \(\Gal(L/\mathbb{Q})\cong\mathbb{Z}/20\).
\(\;(b)\;\) Show that \(L\) contains a unique subfield \(K\) with \([K:\mathbb{Q}]=4\) and explain why \(\Gal(L/K)\cong\mathbb{Z}/5\).
\(\;(c)\;\) Use the Lagrange resolvent construction to find an element \(a\in K\) such that \(L=K(\sqrt[5]{a})\).
Exercise 7.5 Let \(K=\mathbb{Q}(i)\), \(M=\mathbb{Q}(\theta)\) and \(L=\mathbb{Q}(i,\theta)\) where \(\theta=\sqrt{2+\sqrt{2}}\).
\(\;(a)\;\) Prove that \(M/\mathbb{Q}\) is Galois and \(G=\Gal(M/\mathbb{Q})\) has order \(4\).
\(\;(b)\;\) Show that \(G=\langle\,\sigma\,\rangle\cong\mathbb{Z}/4\) where \(\sigma(\theta)=\sqrt{2-\sqrt{2}}\).
\(\;(c)\;\) Prove that there exists \(a\in K\) such that \(L=K(\sqrt[4]{a})\).
\(\;(d)\;\) Apply the Lagrange resolvent construction to find \(a\) as above and deduce that \(L=\mathbb{Q}(\zeta_{16})\) where \(\zeta_{16}\) is a primitive \(16\)-th root of unity.
Exercise 7.6 Let \(K=\mathbb{Q}(\theta)\) where \(\theta=\sqrt{13-2\sqrt{13}}\).
\(\;(a)\;\) Prove that \(K/\mathbb{Q}\) is a Galois extension with \(\Gal(K/\mathbb{Q})\cong\mathbb{Z}/4\).
\(\;(b)\;\) Find \(a\in \mathbb{Q}(i)\) such that \(K(i)=\mathbb{Q}(i,\sqrt[4]{a})\).
Exercise 7.7 Let \(K=\mathbb{F}_2(\omega)\) where \(\omega\) is a root of \(x^2+x+1\in\mathbb{F}_2[x]\) and let \(L=\mathbb{F}_2(\theta)\) where \(\theta\) is a root of \(x^3+x+1\in\mathbb{F}_2[x]\).
\(\;(a)\;\) Explain why there exists \(a\in K\) such that \(L(\omega)=K(\sqrt[3]{a})\).
\(\;(b)\;\) Apply the Lagrange resolvent construction to find suitable \(a\) as above.
Exercise 7.8 Use Kummer Theory to
\(\;(a)\;\) find an irreducible polynomial in \(\mathbb{F}_5[x]\) of degree \(4\),
\(\;(b)\;\) find an irreducible polynomial in \(\mathbb{F}_7[x]\) of degree \(4\),
\(\;(c)\;\) find an irreducible polynomial in \(\mathbb{F}_2[x]\) of degree \(20\).