Read: Read Section 3.1.2.
Turn in: 3.21, 3.22, 3.23, 3.24, 3.27
- When working with complex numbers, addition is easiest in the usual rectangular form \(a +bi\), but multiplication is much easier in the polar form \(r\cos(\theta) + ir\sin(\theta)\) (see Theorem 3.12). So, when exploring powers of \(\zeta_n\), I recommend working in polar form and using DeMoivre’s formula (Corollary 3.14).
- 3.23 should feel very reminiscent of cyclic groups from 110A. You should consider using the division algorithm to prove it.
- 3.24 should follow pretty quickly from the previous problems—you just have to tie them together.
- Use Theorem 3.26 when you work on 3.27 (even though you are not being asked to prove 3.26). To do this, you need to determine one particular root \(b\) of the given polynomial (by “solving for \(x\)”), and then you can apply Theorem 3.26 to get the remaining roots. The first poly should have 3 roots and the second should have 5.
- A lot is presented in this section, but Theorems 3.24 and 3.26 are the key ones to remember. (The other results are just building towards them.)
Extra practice: 3.20, 3.25, 3.26