Stable transcendence for formal power series, generalized Artin-Schreier polynomials and a conjecture concerning p-groups

Let f(x) be a formal power series with coefficients in the field k and let n ? 1. We define the notion of n-transcendence of f(x) over k and, more generally, the stable transcendence function dk(f(x), n). It is shown that, if k has prime characteristic p, this function determines the minimal Krull dimension dk(G) of the universal modular Galois-algebras of an elementary Abelian p-group G, introduced in [2, 3, 4, 5]. Since the concept of n-transcendence is of independent interest in all characteristics, a number of fundamental theorems are proved where the generalized Artin-Schreier polynomials surprisingly play a central role. We make a plausible conjecture in the case when k = Fp, the truth of which would imply a conjectural result concerning dFp (G) previously investigated by the authors.