  We study the general linear groups of infinite fields (or more generally
connected semi-local rings with infinite residue fields) from the perspective
of $E_\infty$-algebras. We prove that there is a vanishing line of slope 2 for
their $E_\infty$-homology, and analyse the groups on this line by determining
all invariant bilinear forms on Steinberg modules. We deduce from this a number
of consequences regarding the unstable homology of general linear groups, in
particular answering questions of Rognes, Suslin, Mirzaii, and others.
