Read: Read end of Chapter 3.
- Make sure to get the updated version of our book! The numbering changed again (starting around 3.60). Sorry again!
Turn in: 3.58, 3.59, 3.65, 3.67, 3.69, 3.70, 3.72
- You’ll want to use 3.56 to prove 3.59.
- You’ll probably prove 3.67 in two steps by separately showing $LHS \subseteq RHS$ and $RHS \subseteq LHS$. There is a hint in the back of the book if you feel stuck.
- You can do 3.70 without actually finding a nice form for the elements of $\mathbb{Q}(\alpha)$, which is quite hard at this point. You just need to work with the definition of $\mathbb{Q}(\alpha)$ (as the smallest subfield of $\mathbb{C}$ that contains both $\mathbb{Q}$ and $\alpha$) and remember that we saw 3.58 that $\{a+b\alpha\mid a,b\in\mathbb{Q}\}$ is not a field for $\alpha = \sqrt{2} + i$.
- For 3.72, you’ll again probably want to separately show $LHS \subseteq RHS$ and $RHS \subseteq LHS$. Theorem 3.71 is helpful, and there is a hint in the book.
Please try (but do not turn in): 3.60, 3.68, 3.71
