Proving Vector Space of F[x]/(g(x)) with Degree n

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SUMMARY

The discussion focuses on proving that the quotient space F[x]/(g(x)) is an n-dimensional vector space, where g(x) is a polynomial in F[x] of degree n. The basis B = (1, x², ..., x^(n-1)) is identified as spanning F[x]/(g(x)). The main challenge presented is demonstrating the linear independence of the basis B within this vector space. The conversation emphasizes the need to relate the basis B to the field F and the polynomial g(x) to establish the proof definitively.

PREREQUISITES
  • Understanding of polynomial rings, specifically F[x]
  • Knowledge of vector space definitions and properties
  • Familiarity with linear independence concepts
  • Basic grasp of quotient structures in algebra
NEXT STEPS
  • Study the properties of polynomial rings and their quotients
  • Learn about linear independence in vector spaces
  • Explore the relationship between bases and dimensions in vector spaces
  • Investigate examples of proving linear independence in polynomial contexts
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Graduate students, algebra enthusiasts, and anyone studying advanced topics in linear algebra and polynomial theory will benefit from this discussion.

johnson123
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Homework Statement



Show that F[x]/( g(x) ) is a n-dimensional vector space. where g is in F[x],
and g has degree n.

Its clear that F[x]/( g(x) ) is a vector space and that

B= (1,[tex]x^{2}[/tex],...,[tex]x^{n-1}[/tex]) spans F[x]/( g(x) ),

but I am having trouble showing that B is linearly independent

I realize this is pretty much a HW problem and it should be in the HW section, but I
read a post from one of the pf mentors noting that for gradlevel/seniorlevel problems
you might have a chance at a response from the non homework sections. thanks for any suggestions.
 
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Well, what happens if they are linearly dependent, so that a nontrivial linear combination of them is equal to zero in F[x] / (g(x))?
 
It's not clear that you have tied B to either F[x] or g(x). First relate B to F and g. Assume for the moment that I am not the person who doesn't have the answer.
 

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