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How many idempotent elements are in Z_m or How many roots have this polynomial in Z_m

f(x)=x^2 +x procedure of proof is important for me.

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closed as too localized by Gjergji Zaimi, Andres Caicedo, Ryan Budney, Torsten Ekedahl, Bruce Westbury Nov 14 '11 at 8:40

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That looks too much like homework: -1. –  Julien Puydt Nov 14 '11 at 6:43
    
maybe you meant x^2-x? Also, by Z_m do you mean Z/mZ or p-adics?(or something else entirely? –  B. Bischof Nov 14 '11 at 6:43

1 Answer 1

Idempotent elements are roots of $g(x)=x^2-x$; my answer will apply equally well to $f(x)=x^2+x$.

The important step is the Chinese Remainder Theorem: one way of stating it is that if $m=p_1^{r_1}\times\cdots\times p_k^{r_k}$ is factored into powers of distinct primes, then the ring $Z_m$ is equal to the direct product of rings $Z_{p_1^{r_1}} \times \cdots \times Z_{p_k^{r_k}}$. So it suffices to count the number of roots in each ring $Z_{p_i^{r_i}}$ and then multiply those numbers together to obtain the number of roots in $Z_m$.

I think it will be easy to convince yourself that the polynomial $g(x)=x^2-x$ has exactly two roots in any ring of the form $Z_{p^r}$.

(By the way, the isomorphism between $Z_m$ and $Z_{p_1^{r_1}} \times \cdots \times Z_{p_k^{r_k}}$ is completely explicit, so this even gives a way to construct the idempotent elements, not just count them.)

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why polynomial g(x)=x 2 −x has exactly has two roots in any ring of the form Z p r ? –  david Nov 14 '11 at 7:05
    
@george martin: please don't answer homework here. –  Martin Brandenburg Nov 14 '11 at 10:00

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