TITLE: Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups
QUESTION [6 upvotes]: I've recently read the following line in an interesting paper:

It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial functions on the Cartan sub algebra $\frak{h}$ by the ideal generated by the fundamental invariants $f_1 , . . . , f_r, r = $rank$({\frak h})$, of the Weyl group W, i.e.
  $$
H^∗(G/B,{Q}) \simeq {\text Sym}_Q{\frak h}^∗/(f_1, . . . , f_r).
$$

I would like to ask:
(1) Does this result extend to other fields, i.e. the complex and real case?
(2) What is a good understandable reference for learning about this result?

REPLY [5 votes]: (1) Assuming you are referring to the coefficient field for your cohomology theory, then yes, the result immediately extends to $\mathbb{R}$ and $\mathbb{C}$ coefficients.
(2) Two sources are Fulton's Young Tableaux (Chapters 9 and 10) and Manivel's Symmetric Functions, Schubert Polynomials, and Degeneracy Loci (Chapter 3).  Both of these sources assume a basic knowledge of and familiarity with algebraic geometry.
Edit: The original source for the result is a paper of Armand 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.