Homework 19

Read: Continue with Chapter 6 up through Section 6.1.1.

• Changes were made to our book. Please download the updated version .

Turn in: 6.9, 6.10, 6.11, 6.13, 6.14, 6.16, 6.17

• Pease read Definition 6.9 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 might be short. You just need to use the def of $F(\alpha)$ and its closure properties.
• For 6.16, most of the work has been done; you just need to connect things. (Look back at 6.8 and 6.14.)
• 6.17 should follow fairly quickly from 6.16 using the ideas in the paragraph before 6.17.

Please try (but do not turn in): 6.12, 6.15, 6.18, 6.19

• 6.19 should follow fairly quickly from 6.17 and 6.18.