Homework 8

Due Date: Friday, April 8

1. 1. Prove that \[ \Phi_{p^n}(x) \,=\, \sum_{k=0}^{p-1} x^{kp^{n-1}} \] for every prime power \(p^n\).
   2. Prove that \[ \Phi_{2m}(x) \,=\, \Phi_m(-x) \] for every odd number \(m > 1\).
2. 1. Let \(\mathbb{F}\) be a field, let \(a,b\in\mathbb{F}^\times\), and suppose that \(2a^2 =1\) and \(\mathrm{ord}_{\mathbb{F}}(b)=4\). Prove that \(\mathrm{ord}_{\mathbb{F}}(a+ab) = 8\).
   2. Use part (a) to find the four elements of order \(8\) in the field \(\mathbb{Z}_{71}(i)\). (Note that this field has \(71^2 = 5041\) elements.)
   3. Let \(p\) be a prime for which \(p \equiv 1\;(\mathrm{mod}\;4)\), and suppose that \(2\) is a quadratic residue modulo \(p\). Use part (a) to prove that \(p \equiv 1\;(\mathrm{mod}\;8)\).
3. 1. Prove that \[ \cos\Bigl(\dfrac{2\pi}{5}\Bigr) \,=\, \dfrac{-1+\sqrt{5}}{4}. \]
   2. Use the identity in part (a) to express \(\sqrt{5}\) as an integer linear combination of the 5th roots of unity.
   3. Let \(p\) be a prime, and suppose that \(p \equiv 1\;(\mathrm{mod}\;5)\). Use your answer to part (b) to prove that \(5\) is a quadratic residue modulo \(p\).