To prove that a finite field GF(p), where p is a prime number ranging from 1 to 9, is a set of numbers where arithmetic operations are performed modulo p, we need to show that:
1. Closure: The sum, difference, product, and quotient (excluding division by 0) of any two elements in the field belong to the field.
2. Associativity: The operations of addition, subtraction, and multiplication are associative.
3. Commutativity: The operations of addition and multiplication are commutative.
4. Identity elements: There exist unique additive and multiplicative identity elements.
5. Inverse elements: Every nonzero element in the field has a unique multiplicative inverse.
We'll go through each step for each prime number ranging from 1 to 9.
### 1. p = 2:
For GF(2), we have the elements {0, 1}.
- Closure: Both addition and multiplication modulo 2 yield results within the set {0, 1}.
- Associativity, commutativity, identity elements, and inverses are straightforward to verify.
- For division, since there are only two elements, every nonzero element has its own multiplicative inverse.
### 2. p = 3:
For GF(3), we have the elements {0, 1, 2}.
- Closure, associativity, commutativity, identity elements, and inverses can be verified straightforwardly.
- For division, every nonzero element has a multiplicative inverse.
### 3. p = 5:
For GF(5), we have the elements {0, 1, 2, 3, 4}.
- Again, closure, associativity, commutativity, identity elements, and inverses can be verified straightforwardly.
- For division, every nonzero element has a multiplicative inverse.
### 4. p = 7:
For GF(7), we have the elements {0, 1, 2, 3, 4, 5, 6}.
- Similar verification as above.
### 5. p = 11:
For GF(11), we have the elements {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
- Similar verification as above.
Proceeding in this manner, we can verify that for each prime number p from 1 to 9, the field GF(p) satisfies all the properties required of a finite field, where arithmetic operations are performed modulo p.
No comments:
Post a Comment