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

• johnson123
In summary, the conversation discusses proving that F[x]/(g(x)) is a n-dimensional vector space, where g is in F[x] and has degree n. It is mentioned that B=(1,x^2,...,x^(n-1)) is a spanning set for F[x]/(g(x)), but there is difficulty in showing that B is linearly independent. Suggestions are given to relate B to F and g in order to prove its linear independence.
johnson123

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,$$x^{2}$$,...,$$x^{n-1}$$) 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.

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.

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of objects, called vectors, and a set of operations that can be performed on these vectors, such as addition and scalar multiplication. The operations follow certain rules, such as closure and associativity, and the set of vectors must satisfy certain properties, such as having a zero vector and being closed under addition and scalar multiplication.

2. What does it mean to prove a vector space?

Proving a vector space means showing that a given set of vectors and operations satisfy all the necessary properties to be considered a vector space. This involves demonstrating that the set is closed under the required operations, follows the necessary rules, and has the required properties.

3. What is F[x]/(g(x))?

F[x]/(g(x)) represents the quotient ring of polynomials in the variable x with coefficients from a field F, by the ideal generated by a polynomial g(x). This means that it consists of all possible polynomials with coefficients from F, but with the additional restriction that g(x) is the zero polynomial.

4. What does it mean to prove a vector space with degree n?

Proving a vector space with degree n means showing that the set of vectors satisfies all the necessary properties to be considered a vector space, while also taking into account the degree of the polynomials involved. This may involve additional steps or restrictions, depending on the specific problem.

5. What are some common methods for proving vector spaces?

Some common methods for proving vector spaces include showing that the set of vectors is closed under the required operations, using mathematical induction to show that the properties hold for all possible vectors, and using properties of fields and rings to demonstrate the required properties. It may also involve using specific techniques or theorems, depending on the specific problem at hand.

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