We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes the rings $\mathbf{Z}/p^n$ where $p$ is a prime. The algebraic description allows us to describe a practical algorithm to compute individual K-groups as well as to obtain several theoretical results: the vanishing of the even K-groups in high degrees, the determination of the orders of the odd K-groups in high degrees, and the degree of nilpotence of $v_1$ acting on the mod $p$ syntomic cohomology of $\mathbf{Z}/p^n$.
@preprint{antieau$K$theory$mathbfZP^n$2024,
title = {On the $K$-theory of $\mathbb{Z}/p^{n}$},
author = {Antieau, Benjamin and Krause, Achim and Nikolaus, Thomas},
year = 2024,
month = may,
number = {arXiv:2405.04329},
eprint = {2405.04329},
primaryclass = {math},
publisher = {arXiv},
doi = {10.48550/arXiv.2405.04329},
}