Understanding the Chinese Remainder Theorem for $\mathbb{Z}^{\times} _{20}$

In summary, the conversation discusses the Chinese Remainder Theorem and its application to proving the isomorphism between $\mathbb{Z}^{\times} _{20}$ and $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$. The theorem states that $\mathbb{Z}^{\times} _{20}$ is isomorphic to $\mathbb{Z}^{\times} _{2^2} \times \mathbb{Z}^{\times} _{5}$, and the participants speculate on the reasons for this isomorphism. The conversation ends with a request for a reference or proof for the statement.
  • #1
NoName3
25
0
How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$?

I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from multiplicative group to additive product like we have here.
 
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  • #2
NoName said:
How do I show that $\mathbb{Z}^{\times} _{20} ≅ \mathbb{Z}_{2} \times \mathbb{Z}_{4}$?

I read that the chinese remainder theorem is the way to go but there are many versions and I can't find the right one. Most versions that I have found are statements between multiplicative groups, not from multiplicative group to additive product like we have here.

Hi NN!

Indeed, the Chinese Remainder Theorem says that:
$$\mathbb Z^\times_{20} \simeq \mathbb Z^\times_{2^2} \times \mathbb Z^\times_{5}$$
Is $\mathbb Z^\times_{2^2}$ isomorphic to $\mathbb{Z}_{2}$? (Wondering)
 
  • #3
I like Serena said:
Hi NN!

Indeed, the Chinese Remainder Theorem says that:
$$\mathbb Z^\times_{20} \simeq \mathbb Z^\times_{2^2} \times \mathbb Z^\times_{5}$$
Is $\mathbb Z^\times_{2^2}$ isomorphic to $\mathbb{Z}_{2}$? (Wondering)
Hi, I like Serena,

Thanks for the reply. Yes, I think they're isomorphic.
 
  • #4
NoName said:
Hi, I like Serena,

Thanks for the reply. Yes, I think they're isomorphic.

So?

Oh, and why do you think they are isomorphic? (Wondering)
 
  • #5
I like Serena said:
So?

Oh, and why do you think they are isomorphic? (Wondering)
So $\mathbb Z^{\times}_{20} \simeq \mathbb Z_{2} \times \mathbb Z^\times_{5}$? As for why, two groups of the same order are isomorphic if $\gcd(n, \phi(n)) = 1$. This satisfies that.
 
  • #6
NoName said:
So $\mathbb Z^{\times}_{20} \simeq \mathbb Z_{2} \times \mathbb Z^\times_{5}$?

Yep!

As for why, two groups of the same order are isomorphic if $\gcd(n, \phi(n)) = 1$. This satisfies that.

Huh? :confused:
I didn't know that yet, but it seems to be true.
Can you provide a reference?
 

1. What is the Chinese Remainder Theorem?

The Chinese Remainder Theorem is a mathematical concept that states that if we have a set of integers that are relatively prime to each other, we can uniquely determine the remainder when dividing any integer by the product of these integers.

2. How is the Chinese Remainder Theorem used in $\mathbb{Z}^{\times} _{20}$?

In $\mathbb{Z}^{\times} _{20}$, the Chinese Remainder Theorem is used to find solutions to equations of the form x ≡ a (mod m), where m is a composite number, and a is any integer. This is useful in solving certain problems related to modular arithmetic.

3. What is the significance of $\mathbb{Z}^{\times} _{20}$ in the Chinese Remainder Theorem?

In the Chinese Remainder Theorem, $\mathbb{Z}^{\times} _{20}$ represents the set of integers coprime to 20. This set is important because the Chinese Remainder Theorem only applies when the moduli are relatively prime to each other.

4. How is the Chinese Remainder Theorem applied in cryptography?

In cryptography, the Chinese Remainder Theorem is used to efficiently encrypt and decrypt messages. The moduli used in the encryption process are usually large prime numbers, and the Chinese Remainder Theorem helps to simplify the decryption process by breaking it down into smaller, simpler equations.

5. Is the Chinese Remainder Theorem limited to just two moduli?

No, the Chinese Remainder Theorem can be extended to any number of moduli, not just two. However, as the number of moduli increases, the equations become more complex and may be more difficult to solve.

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