Reference for integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$ I was looking for a reference which discusses the structure of finite integral extensions of $\mathbb{Z}/p^k\mathbb{Z}$. In particular, I am interested in understanding what the abelian group of its units looks like and is there some Galois/Finite field-like theory for them.
I tried looking for some but can't quite find things directly related to this problem.
Edit: I should mention I can prove things for degree 2 extensions. But for higher degree extensions I can't quite find a complete picture.
 A: As long as $f$ is monic of degree $n$, and irreducible mod $p$, the $\mathbb{Z}/p^k$-algebra $(\mathbb{Z}/p^k)[x]/f(x)$ is flat and has perfect mod $p$ reduction $\mathbb{F}_{p^n}$. The theory of Witt vectors tells you that there is a unique such flat $\mathbb{Z}/p^k$-algebra, which can be identified with $W_{k}(\mathbb{F}_{p^n})=W(\mathbb{F}_{p^n})/p^k$. There is a Galois-theoretic way of thinking about $W(\mathbb{F}_{p^n})$: It is the unique unramified degree $n$-extension of the $p$-adic integers $\mathbb{Z}_p$.
The point is that the ring we get depends only on the mod $p$ reduction of your original polynomial in quite a canonical way.
As soon as you care about $\mathbb{Z}/p^k$-algebras which are not flat, or whose mod $p$ reduction is not perfect, this breaks down and things become messier.
EDIT Regarding units: For any $\mathbb{Z}/p^k$-algebra $A$, we have an exact sequence
$$
1\to (1+pA)\to A^{\times} \to (A/p)^{\times} \to 1,
$$
which is split if $A/p$ is perfect (there is a well-defined map $(A/p)^{\times} \to A^{\times}$, called Teichmüller lift).
In the flat case, where $A= W_k(A/p)$, one can further use the $p$-adic logarithm to identify the multiplicative subgroup $(1+pA)$ with the additive group $W_{k-1}(A/p)$ if $p$ is odd, or with $\{\pm 1\} \times W_{k-2}(A/2)$ is $p=2$. Thus,
$$
W_k(A/p)^{\times} \cong (A/p)^{\times} \times W_{k-1}(A/p) \text{ if $p$ odd},
$$
$$
W_k(A/2)^{\times} \cong (A/2)^{\times}\times \{\pm 1\} \times W_{k-2}(A/2) \text{ if $p=2$},
$$
A: One possibility is the Galois ring $GR(p^k,m)$:
Its additive group is isomorphic to the $m$-fold cartesian product $\mathbb{Z}_{p^k}^m$. However, the latter has no really useful ring structure. The product in the Galois ring $GR(p^k,m)$ mimics that of the Galois field $GF(p^m)$ except that the rules about the primitive element are a bit different, and the coefficients of its powers are integers modulo $p^k$ as opposed to modulo $p$.
The Galois ring $GR(p^k,m)$ is constructed as a quotient ring of the polynomial ring with coefficients from $\mathbb{Z}_{p^k}$
$$
GR(p^k,m):=\mathbb{Z}_{p^k}[x]/\langle f(x)\rangle,
$$
where $f(x)$ is a carefully chosen irreducible monic polynomial of degree $m$ (a Hensel lift of an irreducible polynomial from $\mathbb{Z}_p[x]$).
As a consequence of this we have that we recover the finite field $GF(p^m)$ as a quotient ring of the Galois ring:
$$
GR(p^k,m)/p GR(p^k,m)\simeq GF(p^m).
$$
You can play around with examples of the Galois ring using the online Magma calculator:
http://magma.maths.usyd.edu.au/calc/
The classic book Finite Rings with Identity by McDonald has other rings that may be of interest.
