A finite field, denoted as GF(p), where p is a prime number, is a set of numbers where arithmetic operations like addition, subtraction, multiplication, and division are performed modulo p. Here, we'll consider finite fields GF(p), where p is a prime number ranging from 1 to 9.
1. Finite Field of Order 2 (GF(2)):
This field consists of two elements: {0, 1}. Addition and multiplication are performed modulo 2.
- Addition:
```
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0 (mod 2)
```
- Multiplication:
```
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
```
2. Finite Field of Order 3 (GF(3)):
This field consists of three elements: {0, 1, 2}. Addition and multiplication are performed modulo 3.
- Addition:
```
0 + 0 = 0
0 + 1 = 1
0 + 2 = 2
1 + 0 = 1
1 + 1 = 2 (mod 3)
1 + 2 = 0 (mod 3)
2 + 0 = 2
2 + 1 = 0 (mod 3)
2 + 2 = 1 (mod 3)
```
- Multiplication:
```
0 * 0 = 0
0 * 1 = 0
0 * 2 = 0
1 * 0 = 0
1 * 1 = 1
1 * 2 = 2
2 * 0 = 0
2 * 1 = 2
2 * 2 = 1
```
3. Finite Field of Order 5 (GF(5)):
This field consists of five elements: {0, 1, 2, 3, 4}. Addition and multiplication are performed modulo 5.
- Addition and Multiplication tables can be similarly constructed as above.
4. Finite Field of Order 7 (GF(7)):
This field consists of seven elements: {0, 1, 2, 3, 4, 5, 6}. Addition and multiplication are performed modulo 7.
- Addition and Multiplication tables can be similarly constructed as above.
5. Finite Field of Order 9 (GF(9)):
For the field of order 9, we need to consider a polynomial representation. One way is to use the field extension method, such as GF(3^2) where 3 is a prime and 2 is the degree of the extension.
- We can represent this field using the polynomial representation: GF(3^2) = GF(3)[x] / (x^2 + 1). Here, GF(3) represents the finite field of order 3.
This is a basic overview of finite fields of orders 1 to 9, along with some explanations of how addition and multiplication are performed modulo the prime numbers. For higher order finite fields, more complex representations and operations are used, often involving irreducible polynomials over prime fields.
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