Homework 18

Read: Continue with Chapter 6 up to start of Section 6.1.1.

Turn in: 6.4, 6.9, 6.10, 6.11, 6.13, 6.14

• Fact 6.6 and Definition 6.7 are very important. Please read them carefully.
• In 6.9, you want to find the minimal polynomial for $\zeta_3$ over $\mathbb{Q}$. To do this, you need to find an irreducible polynomial $m(x)$ such that $\zeta_3$ is a root of $m(x)$. Since you know $\zeta_3$ is a root of $x^3-1$, try factoring $x^3-1$ to find an irreducible factor for which $\zeta_3$ is a root.
• 6.14 should be short. You just need to use the definition of $F(\alpha)$ and its closure properties.

Extra practice: 6.12, 6.15