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4. Explain why if p is prime then Z, is the only subdomain of Z,

4. Explain why if p is prime then Z, is the only subdomain of Z,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematics Question:** 4. Explain why if \( p \) is prime then \( \mathbb{Z}_p \) is the only subdomain of \( \mathbb{Z}_p \). **Explanation:** The question asks to explore why the ring of integers modulo a prime number \( p \), denoted as \( \mathbb{Z}_p \), has itself as its only subdomain when \( p \) is a prime number. ### Detailed Explanation: 1. **Definition of \( \mathbb{Z}_p \):** - \( \mathbb{Z}_p \) is the set of integers {0, 1, 2, ..., \( p-1 \)} under addition and multiplication modulo \( p \). - It is a field, which implies every non-zero element has a multiplicative inverse. 2. **Properties of Prime Numbers:** - A prime number \( p \) has exactly two positive divisors: 1 and \( p \). 3. **Subdomains of \( \mathbb{Z}_p \):** - A subdomain of a mathematical structure is a subset that itself satisfies the operations and properties of that structure. - As \( \mathbb{Z}_p \) is a field (due to \( p \) being prime), it has no nontrivial ideals or subdomains. The only subdomains are the trivial ones: the zero ideal and \( \mathbb{Z}_p \) itself. 4. **Conclusion:** - Because fields have no nontrivial subdomains, \( \mathbb{Z}_p \) as a field derived from a prime \( p \) has only \( \{0\} \) and \( \mathbb{Z}_p \) as subdomains. This unique property is critical in abstract algebra and has substantial implications in number theory and cryptography.
Transcribed Image Text:**Mathematics Question:** 4. Explain why if \( p \) is prime then \( \mathbb{Z}_p \) is the only subdomain of \( \mathbb{Z}_p \). **Explanation:** The question asks to explore why the ring of integers modulo a prime number \( p \), denoted as \( \mathbb{Z}_p \), has itself as its only subdomain when \( p \) is a prime number. ### Detailed Explanation: 1. **Definition of \( \mathbb{Z}_p \):** - \( \mathbb{Z}_p \) is the set of integers {0, 1, 2, ..., \( p-1 \)} under addition and multiplication modulo \( p \). - It is a field, which implies every non-zero element has a multiplicative inverse. 2. **Properties of Prime Numbers:** - A prime number \( p \) has exactly two positive divisors: 1 and \( p \). 3. **Subdomains of \( \mathbb{Z}_p \):** - A subdomain of a mathematical structure is a subset that itself satisfies the operations and properties of that structure. - As \( \mathbb{Z}_p \) is a field (due to \( p \) being prime), it has no nontrivial ideals or subdomains. The only subdomains are the trivial ones: the zero ideal and \( \mathbb{Z}_p \) itself. 4. **Conclusion:** - Because fields have no nontrivial subdomains, \( \mathbb{Z}_p \) as a field derived from a prime \( p \) has only \( \{0\} \) and \( \mathbb{Z}_p \) as subdomains. This unique property is critical in abstract algebra and has substantial implications in number theory and cryptography.
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To prove:If p is prime, Zp is an subdomain(internal domain).

 

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