Problem 22 Determine which of the indicated... [FREE SOLUTION]

Chapter 6: Problem 22

Determine which of the indicated rings are fields. $$ \mathbb{Z}_{2}[i]=\left\\{a+b i \mid a, b \in \mathbb{Z}_{2}\right\\} $$

Short Answer

Expert verified

$ \mathbb{Z}_{2}[i] $ is a field as it has no zero divisors and all non-zero elements have inverses.

Step by step solution

Understand the Structure

Identify that $ \mathbb{Z}_{2}[i] $ consists of elements $ a + bi $ where $ a, b \in \mathbb{Z}_{2} $. $ \mathbb{Z}_{2} $ is the field with elements $ \{0, 1\} $. Thus, the elements of $ \mathbb{Z}_{2}[i] $ are $ 0+i0, 0+i1, 1+i0, 1+i1 $. Simplified, these are $ 0, 1, i, 1+i $.

Check for Field Properties

To determine if $ \mathbb{Z}_{2}[i] $ is a field, verify two properties: there must be no zero divisors, and every nonzero element must have a multiplicative inverse.

Check for Zero Divisors

Zero divisors are elements $ a eq 0, b eq 0 $ such that $ ab = 0 $. Verify each non-zero pair: $ i \times 1 = i, \ i \times i = 1 $, and $ (1+i) \times i = 1 $. None of these products are zero, so $ \mathbb{Z}_{2}[i] $ has no zero divisors.

Check for Multiplicative Inverses

Confirm that each non-zero element has a multiplicative inverse in $ \mathbb{Z}_{2}[i] $. Verify: $ 1 $ is its own inverse, $ i \times i = 1 $, and $ (1+i) \times (1+i) = i^2 + 2i + 1 = 1-1+i^2 = 1 $. Each non-zero element has an inverse.

Conclusion: Verify it is a Field

Since $ \mathbb{Z}_{2}[i] $ has no zero divisors, and each non-zero element has an inverse, $ \mathbb{Z}_{2}[i] $ meets the criteria of a field.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Field

A field is a special kind of algebraic structure where you can perform addition, subtraction, multiplication, and division (except by zero) in a consistent way. Fields are foundational in mathematics because they generalize the familiar arithmetic of numbers.

Additive and Multiplicative Inverses: Every element in a field must be able to be "undone" with respect to addition and multiplication.
No Zero Divisors: When you multiply any two non-zero elements, the result cannot be zero.
Associative, Commutative, and Distributive Laws: These laws must hold true for all operations.

Fields can be finite or infinite. For example, the set of real numbers is a field that is infinite, whereas the finite set $ \mathbb{Z}_{2} $ (consisting of integers 0 and 1 under modulo 2 arithmetic) is also a field. By verifying that $ \mathbb{Z}_{2}[i] $ satisfies these conditions, we can conclude it is a field. This makes fields incredibly useful for various applications in mathematics and science.

Zero Divisors

Zero divisors are elements in a ring that can multiply together to give zero, even when neither is zero.

In a true field, there cannot be any zero divisors.
This means that if you're working with a ring and find a zero divisor, you can immediately say it's not a field.

For example, in $ \mathbb{Z}_{2}[i] $, consider the multiplication: $ i \times 1 = i $, $ i \times i = 1 $, and $ (1+i) \times i = 1 $. None of these products result in zero, showing that there are no zero divisors in this ring.Finding zero divisors in a structure is a fundamental step in determining if it's a field or not. The absence of zero divisors in $ \mathbb{Z}_{2}[i] $ supports its classification as a field.

Multiplicative Inverse

The multiplicative inverse of a number in a field is another number that, when multiplied with it, results in one. The concept is crucial to understanding how division works in fields.

Every non-zero element in a field must have a multiplicative inverse.
This is central to a field's operations because it guarantees that division by any non-zero element is possible.

For example, in $ \mathbb{Z}_{2}[i] $, each non-zero element has a multiplicative inverse:

$ i \times i = 1 $
$ 1 \times 1 = 1 $
$ (1+i) \times (1+i) = 1 $

Since all elements have an inverse, this quality confirms $ \mathbb{Z}_{2}[i] $ is indeed a field. Understanding multiplicative inverses highlights how fields allow division to be performed seamlessly.

Complex Numbers over Finite Fields

Complex numbers can also exist over finite fields, such as $ \mathbb{Z}_{2} $. In these fields, every element is a combination of a real component and an imaginary one.

These complex numbers take the form of $ a + bi $, where both $ a $ and $ b $ are elements of the finite field, for instance, $ \mathbb{Z}_{2} $.
The imaginary unit, $ i $, still satisfies $ i^2 = -1 $. However, in $ \mathbb{Z}_{2} $, $ -1 \equiv 1 \mod{2} $, so $ i^2 = 1 $.

The use of finite fields with complex numbers opens up avenues in various mathematical disciplines, including coding theory and cryptography. The structure $ \mathbb{Z}_{2}[i] $ demonstrates how complex numbers behave in these settings, while also fulfilling field criteria like lack of zero divisors and presence of multiplicative inverses. This makes it an interesting case when merging the concepts of complex numbers with finite sets.

91影视

Determine which of the indicated rings are fields. $$ \mathbb{Z}_{2}[i]=\left\\{a+b i \mid a, b \in \mathbb{Z}_{2}\right\\} $$

Short Answer

Step by step solution

Understand the Structure

Check for Field Properties

Check for Zero Divisors

Check for Multiplicative Inverses

Conclusion: Verify it is a Field

Key Concepts

Field

Zero Divisors

Multiplicative Inverse

Complex Numbers over Finite Fields

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