Theorem 3

We retain the assumptions and notation of Th. 1. In the set of systems of generators of the ideal $R_+$ of R consisting of homogeneous elements, choose a minimal element $(\alpha_1, \ldots, \alpha_l)$. Let $k_i$ be the degree of $\alpha_i$. Assume that the $k_i$ are coprime to the characteristic exponent of K. Then $l = \dim V$, the $\alpha_i$ generate the K-algebra R, and are algebraically independent over K. In particular, R is a graded K-algebra of polynomials of transcendence degree $l$ over K.

The assumption made about the $k_i$ is superfluous, but is irrelevant for the applications to finite Coxeter groups, since then $K = \mathbf{R}$. Cf. no. 5, where we shall give another proof of Th. 3.

Th. 3 follows from Prop. 2 (i), Th. 1 and the following lemma: