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C) Prove that Z/5Z is a field.

C) Prove that Z/5Z is a field.

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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I'm sorry, I can't transcribe or interpret parts of the text that are redacted or unclear. However, the visible portion reads: "Prove that \(\mathbb{Z}/5\mathbb{Z}\) is a field." Explanation: In mathematics, proving that \(\mathbb{Z}/5\mathbb{Z}\) is a field involves showing two main properties: 1. **Addition and Multiplication**: These operations are well-defined and follow the usual properties like commutativity, associativity, distributivity, and the existence of an additive identity (0) and a multiplicative identity (1). 2. **Inverses**: Every non-zero element has a multiplicative inverse. Since 5 is prime, every non-zero element in \(\mathbb{Z}/5\mathbb{Z}\) is coprime to 5, thus has an inverse. These properties ensure that \(\mathbb{Z}/5\mathbb{Z}\) is indeed a field.
Transcribed Image Text:I'm sorry, I can't transcribe or interpret parts of the text that are redacted or unclear. However, the visible portion reads: "Prove that \(\mathbb{Z}/5\mathbb{Z}\) is a field." Explanation: In mathematics, proving that \(\mathbb{Z}/5\mathbb{Z}\) is a field involves showing two main properties: 1. **Addition and Multiplication**: These operations are well-defined and follow the usual properties like commutativity, associativity, distributivity, and the existence of an additive identity (0) and a multiplicative identity (1). 2. **Inverses**: Every non-zero element has a multiplicative inverse. Since 5 is prime, every non-zero element in \(\mathbb{Z}/5\mathbb{Z}\) is coprime to 5, thus has an inverse. These properties ensure that \(\mathbb{Z}/5\mathbb{Z}\) is indeed a field.
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