  Let $F$ be a discretely valued complete field with valuation ring
$\mathcal{O}_F$ and perfect residue field $k$ of cohomological dimension $\leq
1$. In this paper, we generalize the Bruhat decomposition in Bruhat and Tits
from the case of simply connected $F$-groups to the case of arbitrary connected
reductive $F$-groups. If $k$ is algebraically closed, Haines and Rapoport
define the Iwahori-Weyl group, and use it to solve this problem. Here we define
the Iwahori-Weyl group in general, and relate our definition of the
Iwahori-Weyl group to that of Haines and Rapoport. Furthermore, we study the
length function on the Iwahori-Weyl group, and use it to determine the number
of points in a Bruhat cell, when $k$ is a finite field.
