TITLE: Find all roots in R[x] of $f(x) = x^2 −2x$ when R = Z/30Z
QUESTION [0 upvotes]: So I'm thinking that for this to be true, we need 
x^2-2x = 0 (mod 5) and x^2-2x = 0 (mod 2) and  x^2-2x = 0 (mod 3)
Then, 
x = 0 (mod 5) or x = 2 (mod 5)
x = 0 (mod 2) or x = 2 (mod 2)
x = 0 (mod 3) or x = 2 (mod 3)
Is this the right approach? Is this how I would represent the roots ?

REPLY [0 votes]: Hint:
Rewrite the equation as $\;(x-1)^2=1$. Thus you have to find the square roots of $1$ modulo $30$. By the Chinese Remainder theorem, this amounts to finding the square roots of $1$ in $\mathbf Z/2\mathbf Z$, in $\mathbf Z/3\mathbf Z$ and in $\mathbf Z/5\mathbf Z$. In a field with characteristic $\ne 2$, it has $2$ square roots, hence you'll obtain $4$ square roots in all.
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