Computing $\operatorname{Aut}(\mathbb{C}[x]/(x-a)^n)$

Question

How do I find $\operatorname{Aut}(\mathbb{C}[x]/(x-a)^n)$ as $\mathbb{C}[x]$ module? Does it have a reasonable description?

Answer 1

I assume you mean $\mathbb C[x]$-automorphisms.

First we will just classify all the endomorphisms, and the identify the automorphisms within there. The module of endomorphisms simply isomorphic to $\mathbb C[x]/(x-a)^n$ itself.

The more general fact is that for a ring $A$ and ideal $I$, the module of endomorphisms of $A/I$ as $A$-module is isomorphic to $A/I$ itself. This is simply because every endomorphism is determined by where it sends $1$ (or rather the class of $1$) and conversely for every such choice one obtains an endomorphism (multiplication by that element).

Under this identification, an endomorphism is an automorphism exactly when it corresponds to multiplication by a unit. The units of $\mathbb C[x]/ (x-a)^n$ are quite easy to classify because it is a local ring with maximal ideal $(x-a)$. The units are the complement of that ideal, which are all elements of the form $$c + g(x)(x-a)^n$$ for $c$ a nonzero complex number.