Proving Zp is a Vector Space for Prime p

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Discussion Overview

The discussion revolves around proving that Zp is a vector space if and only if p is prime. Participants explore the relationship between Zp and vector spaces, as well as the implications of p being prime.

Discussion Character

  • Debate/contested, Conceptual clarification

Main Points Raised

  • One participant questions how to prove that Zp is a vector space if and only if p is prime.
  • Another participant asks for clarification on the structure of Z5 under addition and multiplication, as well as the axioms of vector spaces.
  • Some participants raise the question of what field Zp is a vector space over, suggesting that the problem might actually be about showing Zp is a field if and only if p is prime.
  • One participant asserts that the exercise does not specify a field, implying that any field could be considered, while maintaining that the focus is on vector spaces rather than fields.
  • Another participant states that since any field is a vector space over itself, it suffices to show that Zp is a field if and only if p is prime.

Areas of Agreement / Disagreement

Participants express differing views on whether the discussion should focus on Zp as a vector space or as a field, indicating a lack of consensus on the main objective of the problem.

Contextual Notes

The discussion highlights potential ambiguities in the problem statement regarding the definitions of vector spaces and fields, as well as the assumptions about the field over which Zp is considered.

THEcj39
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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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