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Chinese remainder theorem

  • #1
(a) Let R and S be rings with groups of units R∗ and S ∗ respectively. Prove that
(R × S)∗ = R∗ × S ∗ .
(b) Prove that the group of units of Zn consists of all cosets of k with k coprime to n.
Denote the order of (Zn )∗ by φ(n); this is Euler’s φ-function.
(c) Now suppose that m and n are coprime; prove that φ(mn) = φ(m)φ(n).

I think I know how to do the first one. Let v1 in R and v2 in S, then there exist u1 in R*, u2 in S* such that u1v1=1, u2v2=1. v1 × v2 is in R×S, we have (u1u2)(v1v2)=1, then u1u2 is in (R×S)*. Hence the proof is complete. Is that correct?
for (b), I think the extended euclidean algorithm is helpful here: kK+nN =1, but I'm sure how to come up with a complete proof. Same with the third one.
Could anyone give me some hints here? Any help is greatly appreciated!
 

Answers and Replies

  • #2
352
0
The proof of (a) is not complete. You have proven that [tex]R^\times \times S^\times \subset (R \times S)^\times[/tex], but not the converse.

For (b), your observation is indeed exactly what you need. What does the equation [tex]kK + nN = 1[/tex] look like in [tex]\mathbb{Z}_n[/tex]?

For (c), spend some time looking at what you have done so far.
 

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