6 Problem Sheet 6:
Cyclotomic extensions
Exercise 6.1 Let \(p\) be a prime number and \(r\geq 1\). Show that \(\Phi_{p^r}(x)\) is irreducible over \(\mathbb{Q}\) by applying Eisenstein’s Criterion to \(\Phi_{p^r}(x+1)\).
(Hint: consider the reduction mod \(p\) of \(\Phi_{p^r}(x)\) in \(\mathbb{F}_p[x]\).)
Exercise 6.2 How many monic irreducible factors does \(f(x)=x^{255}-1\in\mathbb{Q}[x]\) have and what are their degrees?
Exercise 6.3 Use Galois Theory to show that \(\sqrt[3]{5}\not\in\mathbb{Q}(\zeta_n)\) for any \(n\geq 1\).
Exercise 6.4 Find all intermediate fields \(\mathbb{Q}\subset M\subset\mathbb{Q}(\zeta_{15})\) with \([M:\mathbb{Q}]=2\).
Exercise 6.5 Let \(K\) be a splitting field of \(f(x)=x^{12}+1\) over \(\mathbb{Q}\). Find the order of \(\Gal(K/\mathbb{Q})\).
Exercise 6.6 Let \(\zeta=\zeta_5=e^{2\pi i/5}\).
\(\;(a)\;\) Show that \(\Gal(\mathbb{Q}(\zeta)/\mathbb{Q})\) is cyclic of order \(4\) with a generator defined by \(\sigma(\zeta)=\zeta^2\).
\(\;(b)\;\) Let \(\theta=\zeta+\zeta^{-1}\). Show that \(K=\mathbb{Q}(\theta)\) is the unique subfield of \(\mathbb{Q}(\zeta)\) with \([K:\mathbb{Q}]=2\).
\(\;(c)\;\) Let \(\Theta=\theta-\sigma(\theta)\). Show that \(\Theta\in K\), \(\Theta^2\in\mathbb{Q}\) and compute the rational number \(\Theta^2\).
\(\;(d)\;\) Prove that \(\theta=\zeta+\zeta^{-1}=(-1+\sqrt{5})/2\) and \(\sigma(\theta)=\zeta^2+\zeta^{-2}=(-1-\sqrt{5})/2\).
\(\;(e)\;\) Determine the minimal polynomial of \(\zeta\) over \(\mathbb{Q}(\sqrt{5})\) and show that \[\zeta=\dfrac{\sqrt{5}-1}{4}+\dfrac{i}{2}\sqrt{\dfrac{5+\sqrt{5}}2}.\]
Exercise 6.7 Let \(\zeta=\zeta_7=e^{2\pi i/7}\).
\(\;(a)\;\) Let \(f(x)\) be the minimal polynomial of \(\theta=\zeta+\zeta^{-1}\) over \(\mathbb{Q}\). Find the degree of \(f(x)\), express the other roots in terms of \(\theta\) and hence find \(f(x)\).
\(\;(b)\;\) Explain why \(\mathbb{Q}(\theta)/\mathbb{Q}\) is a Galois extension and determine its Galois group.
Exercise 6.8 Let \(\zeta=\zeta_{11}=e^{2\pi i/11}\).
\(\;(a)\;\) Show that \(\sigma(\zeta)=\zeta^2\) defines a generator of \(\Gal(\mathbb{Q}(\zeta)/\mathbb{Q})\).
\(\;(b)\;\) Find the degree \([\mathbb{Q}(\alpha):\mathbb{Q}]\) where \(\alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9\in\mathbb{Q}(\zeta)\).
\(\;(c)\;\) Given that \(\displaystyle\Theta=\sum_{j=0}^{9}\,(-1)^j\sigma^j(\zeta)=\sqrt{-11}\), find a simple expression for \(\alpha\).
Exercise 6.9 Let \(\zeta=\zeta_{13}=e^{2\pi i/13}\).
\(\;(a)\;\) Show that a generator of \(\Gal(\mathbb{Q}(\zeta)/\mathbb{Q})\) is given by \(\sigma(\zeta)=\zeta^2\).
\(\;(b)\;\) Explain why \(\mathbb{Q}(\zeta)\) contains unique subfields \(K_2\), \(K_3\), \(K_4\), \(K_6\) with \([K_d:\mathbb{Q}]=d\).
\(\;(c)\;\) For each \(d=2,3,4,6\), prove that \(K_d=\mathbb{Q}(\theta_d)\) where \[\begin{align*} \theta_2&=\zeta+\sigma^2(\zeta)+\sigma^4(\zeta)+\sigma^6(\zeta)+\sigma^8(\zeta)+\sigma^{10}(\zeta), \\ \theta_3&=\zeta+\sigma^3(\zeta)+\sigma^6(\zeta)+\sigma^9(\zeta), \\ \theta_4&=\zeta+\sigma^4(\zeta)+\sigma^8(\zeta), \\ \theta_6&=\zeta+\sigma^6(\zeta). \end{align*}\]
\(\;(d)\;\) Find the minimal polynomial of \(\zeta\) over \(K_6\).
\(\;(e)\;\) Let \(\Theta=\theta_2-\sigma(\theta_2)\). Prove that \(\Theta\in K_2\) and \(\Theta^2\in\mathbb{Q}\). Given that \(\Theta^2=13\), find the minimal polynomial of \(\theta_2\) over \(\mathbb{Q}\).
\(\;(f)\;\) Verify by direct calculation that \((\theta_3-1)\sigma(\theta_3)=-1\) and find the minimal polynomial of \(\theta_3\) over \(\mathbb{Q}\).
Exercise 6.10 \((\star)\) Let \(\zeta=\zeta_{17}=e^{2\pi i/17}\).
\(\;(a)\;\) Show that a generator of \(\Gal(\mathbb{Q}(\zeta)/\mathbb{Q})\) is given by \(\sigma(\zeta)=\zeta^3\).
\(\;(b)\;\) Explain why there is a (unique) tower of field extensions with \([K_d:\mathbb{Q}]=d\): \[\mathbb{Q}\subset K_2\subset K_4\subset K_8\subset \mathbb{Q}(\zeta)\] and prove that \(K_d=\mathbb{Q}(\theta_d)\) where \[\begin{align*} \theta_2&=\zeta+\sigma^2(\zeta)+\sigma^4(\zeta)+\sigma^6(\zeta)+\sigma^8(\zeta)+\sigma^{10}(\zeta)+\sigma^{12}(\zeta)+\sigma^{14}(\zeta), \\ \theta_4&=\zeta+\sigma^4(\zeta)+\sigma^8(\zeta)+\sigma^{12}(\zeta), \\ \theta_8&=\zeta+\sigma^8(\zeta). \end{align*}\]
\(\;(c)\;\) Show that \(\theta_8=2\cos(2\pi/17)\) and
\(\;\;\;\;\;(i)\;\) \(\theta_2\) and \(\sigma(\theta_2)\) are the roots of \(x^2+x-4\in\mathbb{Q}[x]\),
\(\;\;\;\;(ii)\;\) \(\theta_4\) and \(\sigma^2(\theta_4)\) are the roots of \(x^2-\theta_2x-1\in K_2[x]\),
\(\;\;\;(iii)\;\) \(\sigma(\theta_4)\) and \(\sigma^3(\theta_4)\) are the roots of \(x^2-\sigma(\theta_2)x-1\in K_2[x]\),
\(\;\;\;\;(iv)\;\) \(\theta_8\) and \(\sigma^4(\theta_8)\) are the roots of \(x^2-\theta_4x+\sigma(\theta_4)\in K_4[x]\),
\(\;(d)\;\) Solve the quadratic equations to obtain the expression \[\begin{multline*} \qquad\quad\cos\left(\frac{2\pi}{17}\right)=-\frac{1}{16}+\frac{1}{16}\sqrt{17}+\frac{1}{16}\sqrt{34-2\sqrt{17}} \\ +\frac{1}{8}\sqrt{17+3\sqrt{17}-\sqrt{34-2\sqrt{17}}-2\sqrt{34+2\sqrt{17}}}.\qquad\qquad \end{multline*}\]