Read: Read Section 3.2 and start of 3.3.
Turn in: 3.28, 3.33, 3.37, 3.39, 3.40
- On 3.28, your starting assumption is that \(a_nz^n +a_{n-1}z^{n-1} + \cdots + a_2z^2 + a_1z + a_0 = 0\). What happens if take the complex conjugate of both sides of this equation? Fact 3.5 is helpful.
- In problem 3.37, you are asked to prove the subset $U$ of $\mathbb{H}$ is a group. The first thing to address is why $U$ is closed under multiplication from $\mathbb{H}$. Also check that \(U\) contains the identity and is closed under inverses. Additionally, you should note why multiplication is associative, but you won’t really need to prove anything because multiplication is already known to be associative with respect to the larger set $\mathbb{H}$ (and you can just state that).
- In 3.39, the elements of \(S\) are defined to have a particular form. To show closure, take two arbitrary elements of the given form, add/multiply them, and show that the result can be rearranged to have the correct form for $S$.
Extra practice: 3.36, 3.38