at.algebraic topology - Cohomology rings of GL_n(C), SL_n(C)

If $G$ is a connected Lie group (or just a connected loop space with finite homology) then $H^*(G,mathbf{Q})$ is a Hopf algebra. Graded connected Hopf algebras over $mathbf{Q}$ are always tensor products of exterior algebras in odd degrees with polynomial algebras in even degrees. Since polynomial algebras are infinite, they can't occur. The reference to Hopf is probably H. Hopf, Über die algebraische Anzahl von Fixpunkten, Math. Z. 29 (1929), 493–524. For the classification of Hopf algebras, see also A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de
groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207.