Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$.
$((t-a)^n)$ is the ideal generated by $(t-a)^n$.
There is a similar proposition but I am not sure if it is related. Let $R$ be a ring and $f(x)$ be a monic polynomial with coefficients in $R$. Let $R[a]$ denote the ring obtained by adjoining an element satisfying the relation $f(a) = 0$. The powers $1,a,a^2, \dots, a^{n-1}$ form a basis for $R[a]$ over $R$. Note that the elements of $R[a]$ are of the form of a linear combination $r_0 + r_1a + ...r_{n-1}a^{n-1}$ where each $r_i\in R$
I know that $\mathbb{C}$ is a ring.