# Proving Vector Space Dimensionality of F[x]/(g(x))

• johnson123
In summary, the problem is to show that F[x]/( g(x) ) is a n-dimensional vector space, where g is a polynomial of degree n in F[x]. It is clear that B=(1,x^{2},...,x^{n-1}) spans F[x]/( g(x) ), but the difficulty lies in proving that B is linearly independent. To do so, a matrix can be constructed with B as the first row and each subsequent row being the derivative of the previous row. If the determinant of this matrix is zero, then it implies that the columns of the matrix are linearly dependent and therefore, B is not linearly independent. The determinant can be evaluated using the field's characteristic, if it is
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

Write a matrix whose first row is B=(1,x,x^2...x^(n-1)), whose second row is the first derivative of the first, whose third row is the second derivative of the first etc. If B were linearly independent, then the columns of the matrix would be linearly dependent, so the determinant would be zero. Now evaluate the determinant. You may have to do some extra head scratching if the characteristic of your field isn't zero.

It shouldn't matter what the characteristic of the field is. Just write down a relation among the x^k with coefficients from F. If this is 0 in F[x] / g(x), then it means that it lives in the ideal g(x), i.e., is a polynomial times g(x). I leave the rest to you.

## What is a vector space and how is it related to F[x]/(g(x))?

A vector space is a mathematical structure that consists of a set of elements, called vectors, and two operations, addition and scalar multiplication, that satisfy certain properties. F[x]/(g(x)) is the notation for the quotient ring formed by the polynomial ring F[x] and the ideal generated by g(x), which is a way of representing the set of all polynomials in F[x] with coefficients modulo g(x).

## What does it mean to prove the dimensionality of a vector space?

To prove the dimensionality of a vector space means to show that the space has a specific number of independent vectors, known as the dimension, that can be used to generate all other vectors in the space through linear combinations. This is an important concept in linear algebra and is often used to understand and analyze the properties of vector spaces.

## What methods can be used to prove the dimensionality of a vector space?

There are several methods that can be used to prove the dimensionality of a vector space, including the use of basis and spanning set, linear independence, and the rank-nullity theorem. In the case of F[x]/(g(x)), the most common method is to use the basis and spanning set approach, which involves finding a set of independent vectors that span the entire space.

## How does the degree of g(x) affect the dimensionality of F[x]/(g(x))?

The degree of g(x) plays a crucial role in determining the dimensionality of F[x]/(g(x)). In general, the dimension of the quotient space will be equal to the degree of g(x). This means that the higher the degree of g(x), the larger the dimension of the quotient space will be. However, there are exceptions to this rule, and the dimensionality may also depend on the specific properties of g(x) and the underlying field F.

## Can the dimensionality of F[x]/(g(x)) change under different choices of g(x)?

Yes, the dimensionality of F[x]/(g(x)) can change under different choices of g(x). This is because the dimensionality is directly related to the degree of g(x), and different polynomials can have different degrees. Additionally, the dimensionality may also be affected by the specific properties of g(x) and the underlying field F. Therefore, it is important to carefully consider the choice of g(x) when proving the dimensionality of F[x]/(g(x)).

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