MHB Proving Zp is a Vector Space for Prime p

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Zp can be proven to be a vector space if and only if p is prime, as it must also be a field under addition and multiplication. The discussion emphasizes that the axioms of a vector space are satisfied when Zp is treated over a suitable field. There is some confusion regarding whether the task is to demonstrate Zp as a field instead of a vector space, but the consensus is that the focus remains on vector spaces. The exercise does not specify the field, but it is noted that any field can serve as a vector space over itself. Ultimately, the key takeaway is that Zp's status as a vector space hinges on p being prime.
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How can I prove that
Zp is a vector space if and only if p is prime
 
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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.
 

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