Chern character on an even-dimensional sphere is integral

Under the natural identificationthe Chern character maps ${\tilde{K}}^0(S^{2n})$ into integral cohomology. A Bott generator is an exterior product of $n$ degree-two Bott elements, whose Chern characters multiply to an integral top-dimensional generator.

For a complex vector bundle $E\to S^{2n}$,is divisible by $(n-1)!$. Since the lower Chern classes vanish, the Newton identity givesand the Chern character on an even-dimensional sphere is integral.