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

[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

 

[Dick]

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.

 

[masnevets]

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.