The mod 2 Homology of the General Linear Group of a 2-adic Local Field

Let F be a finite extension of the 2-adic rational numbers. We compute the mod 2 homology of the general linear group GL(F) as a Hopf algebra over the Steenrod algebra. The answer is formulated in terms of the well-known homology algebras of the infinite unitary group U, its classifying space BU, and the classifying space BO of the infinite orthogonal group.