Proving Zp is a Vector Space for Prime p

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SUMMARY

Zp is a vector space if and only if p is a prime number. The discussion emphasizes that Zp, under addition and multiplication, adheres to the axioms of a vector space when p is prime. The participants clarify that while Zp can be viewed as a field, the primary focus is on its properties as a vector space. The relationship between Zp being a vector space and the primality of p is established as a fundamental concept in linear algebra.

PREREQUISITES
  • Understanding of vector space axioms
  • Knowledge of fields in abstract algebra
  • Familiarity with modular arithmetic, specifically Zp
  • Basic concepts of linear transformations
NEXT STEPS
  • Study the properties of vector spaces over finite fields
  • Learn about the structure of Zp as a field when p is prime
  • Explore the implications of vector space axioms in different mathematical contexts
  • Investigate linear transformations and their applications in vector spaces
USEFUL FOR

Mathematics students, educators, and researchers interested in abstract algebra, particularly those focusing on vector spaces and fields.

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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