Polynomial Rings: Finding 8 Elements with r^2=r

[jgens]

Homework Statement

Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r.

Homework Equations

N/A

The Attempt at a Solution

The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a technique for finding a few more elements?

Thanks!

 

[micromass]

You can always brute force it. An arbitrary element in your ring has the form [ax^3+bx^2+cx+d]. Square it and see when the relation is satisfied.