The $\mathbb Z_p$-rank of a topological $K$-group

A complete two-dimensional local field $K$ of mixed characteristic with finite second residue field is considered. It is shown that the rank of the quotient $U(1)K_2^{\mathrm{top}}K/T_K$, where $T_K$ is the closure of the torsion subgroup, is equal to the degree of the constant subfield of $K$ over $\mathbb Q_p$. Also, a basis of this quotient is constructed in the case where there exists a standard field $L$ containing $K$ such that $L/K$ is an unramified extension.