# Math Help - Solving Equations in a factor Ring Z[x] / <x³>

1. ## Solving Equations in a factor Ring Z[x] / <x³>

How do I solve y² -2y + 1 = 0 in the factor ring Z[x] / <x³> ?
I know how to solve this in Z or Z_n (where n is small), but I'm not too familiar with factor rings.

How do I solve y² -2y + 1 = 0 in the factor ring Z[x] / <x³> ?
I know how to solve this in Z or Z_n (where n is small), but I'm not too familiar with factor rings.

In any unitary ring, $y^2-2y+1=(y-1)^2$ , so...

Tonio

3. I know that y = 1 is one solution, but are there others? Like in Z_n, there can be other solutions than just y = 1.

You know that $(y-1)^2=0$. You now need to decide whether there are any nonzero elements in this ring whose square is 0. If not, then you can conclude that $y-1=0.$