Topological Hochschild Homology of $K\mathrm{/}p$ as a $K_p^{\wedge }$ module

Let $R$ be an $E_{\mathrm{\infty }}$-ring spectrum. Given a map $\zeta$ from a space $X$ to $BG{L}_{1}R$, one can construct a Thom spectrum, $X^{\zeta }$, which generalises the classical notion of Thom spectrum for spherical fibrations in the case $R=S^0$, the sphere spectrum. If $X$ is a loop space ($\simeq \mathrm{\Omega }Y$) and $\zeta$ is homotopy equivalent to $\mathrm{\Omega }f$ for a map $f$ from $Y$ to $B^{2}G{L}_{1}R$, then the Thom spectrum has an $A_{\mathrm{\infty }}$-ring structure. The Topological Hochschild Homology of these $A_{\mathrm{\infty }}$-ring spectra is equivalent to the Thom spectrum of a map out of the free loop space of $Y$. This paper considers the case $X=S^1$, $R=K_p^{\wedge }$, the p-adic $K$-theory spectrum, and $\zeta =1-p\in {\pi }_{1}BG{L}_1{K}_p^{\wedge }$. The associated Thom spectrum $(S^1{)}^{\zeta }$ is equivalent to the mod p $K$-theory spectrum $K\mathrm{/}p$. The map $\zeta$ is homotopy equivalent to a loop map, so the Thom spectrum has an $A_{\mathrm{\infty }}$-ring structure. I will compute ${\pi }_{\ast }TH{H}^{K_p^{\wedge }}(K\mathrm{/}p)$ using its description as a Thom spectrum.