It is known that there is a bijection between the set of characteristic classes of rank $k$ complex vector bundles and the ring of symmetric formal power series in $k$ variables, given by identifying elements of the cohomology ring $H^*(G_k(\mathbb{C}^\infty))$ of the $k$-th Grassmannian with symmetric polynomials in $x_1,...,x_k$, where each $x_i$ is a generator of $H^*(G_1(\mathbb{C}^\infty))$.
I'd like to know: what is the coefficient ring (I'm guessing $\mathbb{R}$ or $\mathbb{C}$) over which we are defining these formal power series for this bijection to hold? The reason for my question is that I have only ever seen characteristic classes that are represented by coefficients in $\mathbb{Q}$. Are the other series somehow less interesting?
You can take coefficients in any commutative ring. This follows from a computation of the integral homology and cohomology together with the universal coefficient theorem. In practice the particular power series people care about (e.g. the Todd class, the Chern character) end up being defined over $\mathbb{Q}$.