Show Q[x]/(x^2-4x+4) has infinite nilpotent elements

[Rederick]

Homework Statement

Let R = Q[x]/(x^2-4x+4). Show that R has infinitely many nilpotent elements.

Homework Equations

The Attempt at a Solution

I see that x^2-4x+4 = (x-2)(x-2) and that a nilpotent means a^2 = 0. Other than that, I'm not sure where to start.

 

[micromass]

Let $$\varphi:\mathbb{Q}[x]\rightarrow \mathbb{Q}[x]/(x^2-4x+4)$$ be the quotient map. Can you show that $$\varphi(x-2)$$ is nilpotent? Can you in some way use this to obtain more nilpotents?

 

[Rederick]

Thanks for the direction, I think I got something that will work.