The $l$-adic $K$-theory of a $p$-local field

We verify a special case of a conjecture of G. Carlsson that describes the $\text{l}$-adic $K$-theory of a field $F$ of characteristic prime to $\text{l}$ in terms of the representation theory of the absolute Galois group $G_F$. This conjecture is known to hold in two cases; in this article we examine the second case, in which the field in question is the field of Laurent series over a finite field ${\text{FF}}_p((x))$.