Rational cohomology of an integral Eilenberg–MacLane space ($H^{\ast }(K(\mathbb{Z},n);\mathbb{Q})$)

The ring is exterior on its degree-$n$ universal class for odd $n$, and polynomial for even $n$. The path-loop Serre spectral sequence alternates these two types by transgression. In the polynomial-fiber case its derivation has nonzero coefficients $k$ on powers, which become invertible over the rationals.