Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 16: Problem 6
(a) Prove that if \(p\) and \(q\) are prime, then \(\mathbb{Z}_{p} \times \mathbb{Z}_{q}\) is never a field. (b) Can \(\mathbb{Z}_{p}{ }^{n}\) be a field for any prime \(p\) and any positive integer \(n \geq 2 ?\)
Short Answer
Expert verified
(a) \(\mathbb{Z}_p \times \mathbb{Z}_q\) is not a field as not every non-zero element has an inverse. (b) \(\mathbb{Z}_p^n\) is not a field for \(n\geq 2\) due to zero divisors.
Step by step solution
01
Understand the Definition of a Field
A field is a set equipped with two operations, addition and multiplication, such that it satisfies the field axioms: commutativity, associativity, distributivity, existence of an additive identity and a multiplicative identity, existence of additive inverses, and existence of multiplicative inverses for all non-zero elements.
02
Analyze the Structure of \(\mathbb{Z}_p\)
The set \(\mathbb{Z}_p\), integers modulo \(p\), for a prime \(p\), forms a field. This is because every non-zero element has a multiplicative inverse, as the greatest common divisor of any non-zero element and \(p\) is 1. This satisfies all field conditions.
03
Examine the Structure of \(\mathbb{Z}_p \times \mathbb{Z}_q\)
The Cartesian product \(\mathbb{Z}_p \times \mathbb{Z}_q\) forms a ring but not a field. In a field, every non-zero element must have a multiplicative inverse, but in \(\mathbb{Z}_p \times \mathbb{Z}_q\), an element like \((1,0)\) or \((0,1)\) does not have a multiplicative inverse except when both elements are non-zero, therefore \(\mathbb{Z}_p \times \mathbb{Z}_q\) cannot be a field.
04
Check if \(\mathbb{Z}_p^n\) can be a field
\(\mathbb{Z}_p^n\) refers to the \(n\)-dimensional vector space over the field \(\mathbb{Z}_p\). Its elements are \(n\)-tuples, and the space is not a field when \(n\geq 2\). In a field, there can be no zero divisors; however, in \(\mathbb{Z}_p^n\), one can find non-zero vectors whose dot product is zero, indicating the presence of zero divisors and hence \(\mathbb{Z}_p^n\) cannot be a field for \(n\geq 2\).
05
Conclusion: Compilation of Results
For part (a), \(\mathbb{Z}_p \times \mathbb{Z}_q\) is not a field as it does not satisfy the condition that every non-zero element has an inverse. For part (b), \(\mathbb{Z}_p^n\) is not a field for \(n\geq 2\) because the presence of zero divisors means it fails the field definition.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Prime Numbers
Prime numbers are the building blocks of numbers. A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. In other words, it can only be divided by 1 and the number itself without leaving a remainder. This makes prime numbers fundamental in number theory.
- Examples of prime numbers include 2, 3, 5, 7, and 11. Notice that 2 is the only even prime number.
- Prime numbers are used in various areas of mathematics and play a crucial role in fields like cryptography.
- Every positive integer greater than 1 can be uniquely factored into prime numbers. This is known as the fundamental theorem of arithmetic.
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value, called the modulus. This is akin to the face of a clock, where the numbers reset after reaching 12.
- The notation \( a \equiv b \mod m \) signifies that \( a \) and \( b \) leave the same remainder when divided by \( m \).
- In modular arithmetic, we focus on remainders rather than quotients. This makes it very useful in computer science and cryptography.
- For example, with a modulus of 5, the number 23 is congruent to 3 because when 23 is divided by 5, the remainder is 3.
Vector Spaces
A vector space is a collection of objects called vectors, which can be added together and multiplied by scalars (which are often considered elements of a field). Vector spaces are fundamental in areas such as linear algebra and geometry.
- Vectors in a vector space can be represented as coordinates such as \( (a, b) \) or \( (x_1, x_2, \ldots, x_n) \), where each element represents a dimension or a component.
- An \( n \)-dimensional vector space over a field \( \mathbb{Z}_p \) consists of tuples where operations are performed component-wise, according to the rules of modular arithmetic.
- Even though vector spaces can have properties resembling those of fields, not all vector spaces are fields. The presence of zero divisors in \( \mathbb{Z}_p^n \) for \( n \geq 2 \) prevents it from being a field.
Ring Theory
Ring theory is a branch of algebra that studies rings, which are sets equipped with two binary operations: addition and multiplication. A ring must satisfy certain properties, such as associativity and distributivity, but unlike fields, elements in a ring do not necessarily have multiplicative inverses.
- A ring can be commutative, where multiplication between any two elements is independent of their ordering, or non-commutative.
- In fields, every non-zero element must have an inverse, but a ring may have elements like zero divisors that prevent the structure from being a field.
- The product \( \mathbb{Z}_p \times \mathbb{Z}_q \) provides an example of a ring structure which cannot be a field because elements like \((1,0)\) lack a multiplicative inverse.
