# Proving Zp is a Vector Space for Prime p

• MHB
• THEcj39
In summary, the conversation discusses the proof that Zp is a vector space if and only if p is prime. The concept of vector space and its axioms are also mentioned, along with the question of which field the vector space is over. It is clarified that the problem is to show that Zp is a vector space, not a field. It is stated that any field can be a vector space over itself, but it is sufficient to prove that Zp is a field if and only if p is prime.
THEcj39
How can I prove that
Zp is a vector space if and only if p is prime

What does the Z5 under addition and multiplication look like? What are the axioms of vector space?

A vector space over what field? Are you sure the problem is not to show that Zp is a field if and only if p prime?

Country Boy said:
A vector space over what field? Are you sure the problem is not to show that Zp is a field if and only if p prime?
The exercise doesn't specifies so I think is any field, and yes I'm sure the problem is vector spaces not fields

Any field is a vector space over itself so it is sufficient to show that Zp is a field if and only if p is prime.

## 1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects (vectors) that can be added together and multiplied by scalars (numbers), satisfying certain axioms such as closure, associativity, and distributivity.

## 2. How do you prove that Zp is a vector space?

To prove that Zp (the set of integers modulo p) is a vector space, we need to show that it satisfies the vector space axioms. This includes showing that addition and scalar multiplication are closed operations, that they are associative and commutative, and that there exists a zero vector and additive inverses for every vector.

## 3. What is the significance of p being prime in this proof?

The prime number p is significant because it ensures that the integers modulo p form a field. This means that every nonzero element has a multiplicative inverse, which is necessary for scalar multiplication to be well-defined in a vector space.

## 4. Can you provide an example of a vector in Zp?

Yes, for example, in Z5 (the set of integers modulo 5), the vector [3] would represent the integer 3. This vector can be added to other vectors in Z5, such as [2], to get [3+2]=[0].

## 5. What other properties of Zp can be proven using this vector space structure?

Using the vector space structure of Zp, we can also prove properties such as the existence of a basis (a set of linearly independent vectors that span the space), the dimension of the space (which is equal to p), and the existence of a linear transformation from Zp to itself.

Replies
8
Views
661
Replies
18
Views
1K
Replies
4
Views
5K
Replies
2
Views
6K
Replies
5
Views
1K
Replies
3
Views
360
Replies
7
Views
1K
Replies
6
Views
2K
Replies
11
Views
1K
Replies
4
Views
2K