. Prove that Zp (p is a prime number) is a field

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karton · Accepted Answer

To prove that Zp (where p is a prime number) is a field, we need to show that it satisfies all the axioms of a field. A field is a mathematical structure with two operations, addition and multiplication, that satisfies several properties.
Let's go through the properties one by one:
1. Closure under addition and multiplication:
For any two elements a and b in Zp, their sum (a + b) and product (a * b) are also in Zp. Since Zp consists of integers modulo p, both addition and multiplication will result in remainders modulo p.
2. Associativity of addition and multiplication:
Addition and multiplication in Zp are both associative. For any three elements a, b, and c in Zp, we have:

(a + b) + c = a + (b + c)

(a * b) * c = a * (b * c)

3. Existence of additive and multiplicative identities:
The identity element for addition in Zp is 0. For any element a in Zp, we have:

a + 0 = a

The identity element for multiplication in Zp is 1. For any element a in Zp, we have:

a * 1 = a

4. Existence of additive inverses:
For any element a in Zp, there exists an additive inverse -a in Zp such that:

a + (- a) = 0

5. Existence of multiplicative inverses:
For any non-zero element a in Zp, there exists a multiplicative inverse a^(-1) in Zp such that:

a * a^{(} - 1) = 1

To prove the existence of multiplicative inverses, we can use Fermat's Little Theorem. Since p is a prime number, for any non-zero element a in Zp, we have:

a^{(} p - 1) \equiv 1 (m o d p)

This implies that

a * a^{(} p - 2) \equiv 1 (m o d p)

. Therefore,

a^{(} - 1) = a^{(} p - 2)

is the multiplicative inverse of a in Zp.
By satisfying all the properties of a field, Zp (where p is a prime number) is indeed a field.

. Prove that Zp (p is a prime

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