[SOLVED] group theory problem

ehrenfest

1. The problem statement, all variables and given/known data
Find all solutions of the equation x^3-2x^2-3x=0 in Z_12.


2. Relevant equations



3. The attempt at a solution
We first factor the polynomial into x(x-3)(x+1)=0. Recall that Z_12 is not an integral domain since 12 is not prime (e.g. 3*4=0). Therefore setting each factor equal to 0 WILL NOT GIVE ALL OF THE SOLUTIONS.

Obviously, the solutions to x=0, (x-3)=0, (x+1)=0, x(x-3)=0, x(x+1) = 0, (x-3)(x+1)=0 will also be solutions to our equation. I can find all of those. The problem is that I do not know how to find the remaining ones.

Hurkyl

You could narrow things down by factoring 12 into prime powers, and using the chinese remainder theorem.

You can narrow things down even further in Z/4Z by first considering it in Z/2Z.


Or... you could apply the fact that each solutions will make at least one of the factors a zero divisor..


But honestly, 12 is so small that I'd expect simply trying all 12 possibilities is the most efficient way to find the roots.

ehrenfest

What are the twelve possibilities?

Dick

x=0,1,2...11. What else??

HallsofIvy

From what you said before, it would appear that you know what Z₁₂ is! The" 12 possibilities" Hurkyl mentioned are the 12 elements of that ring.

ehrenfest

Grrrr. Someday I will stop making mistakes like this.