# Determining whether the quotient space of the polynomials is finite.

I'm just a little unsure whether my answers to this question are right.

Let $V=F[x]$ be the vector space of polynomials over the field $F$. Determine whether or not $V/M$ is finite dimensional when $M$ is

i) The subspace $V_{n}$ of polynomials of degree less than or equal to $n$

I've said a basis for this is $(x^{n+1}+M, x^{n+2}+M, ...)$ which is infite, so $V/M$ is not finite dimensional.

ii) The subspace of even polynomials.

I've said a basis for this is $(x+M, x^{3}+M,... , x^{2n+1},...)$ which is infinite, so $V/M$ is not finite dimensional.

iii) The subspace of all polynomials divisible by $x^{n}$

I've said a basis for this is $(1+M, x+M, x^{2}+M,... x^{n-1}+M)$ which is finite, so $V/M$ is finite dimensional.

Are these correct? I feel most unsure about i)

• I thin k this is all fine – Hagen von Eitzen Oct 25 '13 at 15:17
• I concur, it's all fine. – Daniel Fischer Oct 25 '13 at 15:22
• Brilliant. Thanks! – CBC Oct 25 '13 at 16:55