Let $\mathbb{F}[x]$ be the ring polynomials in one variable $x$ over a field $\mathbb{F}$ with the relation $x^n=0$

Question

Let $\mathbb{F}[x]$ be the ring polynomials in one variable $x$ over a field $\mathbb{F}$ with the relation $x^n=0$,for a fixed $n$,is a natural number.Then What is dimension of $\mathbb{F}[x]$ over $\mathbb{F}$?

a) $1$

b) $n-1$

c) $n$

d) $\infty$

Answer 1

Let $g$ be a polynomial in $\mathbb{F}[x]$ then you can write

$$g(x) = \sum_{j=0}^m a_j x^j = x^n\sum_{j \geq n} a_j x^{j-n} + \sum_{j<n}a_j x^j$$

So the polinomials with the above relations have degree at most $n-1$. You can solve your exercise now?

Answer 2

The other answer is nice and well thought gives you the answer. Here you have another approach:

Prove that $\;\{1,\,x,\,x^2,\,...,\,x^{n-1}\}\;$ is a basis of $\;f[x]\;$ .