Read: Read through end of Chapter 3.
Turn in: 3.67, 3.69, 3.70
- 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 $\alpha$, meaning that if some subfield contains both $\mathbb{Q}$ and $\alpha$ then the subfield necessarily contains $\mathbb{Q}(\alpha)$). Also, remember that we saw in 3.58 that \(\{a+b\alpha\mid a,b\in\mathbb{Q}\}\) is not a field for $\alpha = \sqrt{2} + i$.
Extra practice: 3.65, 3.68