Solved: Find Solutions for Group Theory Problem in Z_12

[ehrenfest]

[SOLVED] group theory problem

Homework Statement

Find all solutions of the equation x^3-2x^2-3x=0 in Z_12.

Homework Equations

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]

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

What are the twelve possibilities?

 

[Dick]

What are the twelve possibilities?

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.