What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ?

Question

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ? Can someone please give some links , articles where I can study about polynomila rings and its quotient rings ?

Answer 1

Let $I = \langle x^3 - x\rangle$. For $p \in \Bbb R[x]$, denote the element $p(x) + I$ by $p(t)$.

By the division algorithm, note that every element of $\Bbb R[x]/I$ can be written as $p(t)$ where $p$ is a polynomial of degree at most $2$. It follows that the set $$ B = \{1,t,t^2\} $$ is a spanning set of $\Bbb R[x]/I$. Verify that $B$ is additionally linearly independent, so that it forms a basis.

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Illustrative example: $$ t^5 + t^4 = \\ (t^2+t)(t^3 - t) + t^2 + t = \\ (t^2 + t)\cdot(0) + t^2 + t =\\ t^2 + t = \\ 1 \cdot t^2 + 1 \cdot t + 0 \cdot 1 $$ Intuitively, you can think of $\Bbb R[x]/I$ as $\Bbb R$ extended by a symbol $t$ that we've defined to satisfy $t^3 = t$. Note that, since $x^3 - x$ has zeros in $\Bbb R$, this quotient ring is not a field (since it is not a domain, since it has zero divisors).