# Consider Z4 ({0, 1, 2, 3} mod 4) and GF (4) (also known as GF(2^2)).

• krispiekr3am
In summary, Consider Z4 ({0, 1, 2, 3} mod 4) is a set of integers divided by 4 with the remainder taken, including the numbers 0, 1, 2, and 3. GF(4) is a Galois Field of order 4, a mathematical structure used in coding theory, cryptography, and other areas. The difference between Z4 and GF(4) is that GF(4) is a finite field built on the set Z4. GF(4) is closely related to binary numbers as it is a finite field of size 2^2. It has various applications in mathematics and computer science, including coding theory, cryptography, and data storage
krispiekr3am
(a) Is (Z4, +) a group? Is (Z4, +, *) a ring? Explain.
(b) Is Z4 a field, in other words, does every integer in Z4 have a multiplicative inverse?
(c) Generate the addition table and multiplication table of GF(4).

can someone help me. i am clueless?

Have you had any thoughts on the problem?

In particular, have you written out the addition and multiplication tables for Z4?

## 1. What is Consider Z4 ({0, 1, 2, 3} mod 4)?

Consider Z4 ({0, 1, 2, 3} mod 4) refers to a set of integers modulo 4, which means that the integers are divided by 4 and the remainder is taken. In this case, the set includes the numbers 0, 1, 2, and 3.

## 2. What is GF(4)?

GF(4) stands for Galois Field of order 4, also known as finite field of size 4. It is a mathematical structure used in coding theory, cryptography, and other areas of mathematics.

## 3. What is the difference between Z4 and GF(4)?

Z4 is a set of integers modulo 4, while GF(4) is a mathematical structure with addition and multiplication operations defined over the elements 0, 1, α, and α^2, where α is a primitive element. In other words, GF(4) is a finite field built on the set Z4.

## 4. How is GF(4) related to binary numbers?

GF(4) is closely related to binary numbers because it is a finite field of size 2^2. This means that the elements of GF(4) can be represented using two binary digits, 0 and 1.

## 5. What are the applications of GF(4)?

GF(4) has various applications in mathematics and computer science, such as coding theory, cryptography, and error-correcting codes. It is also used in the design of digital circuits and data storage systems.

• Linear and Abstract Algebra
Replies
7
Views
3K
• Linear and Abstract Algebra
Replies
17
Views
4K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Precalculus Mathematics Homework Help
Replies
27
Views
2K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
3
Views
929
• Linear and Abstract Algebra
Replies
6
Views
2K
• Linear and Abstract Algebra
Replies
1
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
5
Views
3K