  In this paper, we use trace methods to study the algebraic $K$-theory of
rings of the form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute the
relative $p$-adic $K$ groups for $R$ a perfectoid ring. In particular, we get
the integral $K$ groups when $R$ is a finite field, and the integral relative
$K$ groups $K_*(R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2, (x_1,\ldots, x_d))$
when $R$ is a perfect $\mathbb{F}_p$-algebra. We conclude the paper with some
other notable computations, including some rings which are not quite of the
above form.
