**Chapter 1**

**2. **Show that is a cube root of 1 (meaning that its cube equals 1).

Solution:

**3. **Prove that for every .

Solution:

by proposition 1.6

by proposition 1.6

This is the alternate proof suggested in the solutions manual.

**4. **Prove that if and , then or .

Solution:

by proposition 1.5

Now let , then done.

Check: The solution manual provides a different proof

**5.(a)** Determine if is a subspace of .

Solution:

Let

Additive identity in .

Let and

Then

and

Closed under addition.

Let and

Then

Closed under scalar multiplication.