Dartmouth College

Mathematics 81

Homework assigned Wednesday, January 17

1. Let $$\zeta = e^{2\pi i/8}$$ be a primitive eighth root of unity.
   1. Show that $(\zeta + \zeta^{-1})^2 = 2$.
   2. Show that ${\mathbb{Q}}(\sqrt 2) \subseteq {\mathbb{Q}}(\zeta)$.
   3. Compute the degree $[{\mathbb{Q}}(\zeta) : {\mathbb{Q}}(\sqrt 2)]$.

2. Show that $x^3 - 2$ is irreducible over ${\mathbb{Q}}(i)$, $i = \sqrt{-1} \in {\mathbb{C}}$. Do not attempt to factor the polynomial; argue via field extensions.

3. Let $m_1, m_2, \dots, m_t$ be integers.
   1. Show that $[{\mathbb{Q}}(\sqrt{m_1}, \sqrt{m_2}, \dots, \sqrt{m_t}) : {\mathbb{Q}}] \le 2^t$.
   2. Give an example to show the inequality can be strict, and justify by computing degrees.
   3. Now assume the the integers $m_i$ are square-free and are coprime in pairs. Show that $[{\mathbb{Q}}(\sqrt{m_1}, \sqrt{m_2}, \dots, \sqrt{m_t}) : {\mathbb{Q}}] = 2^t$. Hint: Induction on $t$. You proably want to work out the case $t=2$ carefully before trying the general argument.