Arithmetic in generalized Artin-Schreier extensions of k(x)

AbstractIf k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation yp • y = β generates a tower of extensions through Ki = Ki−1(yi) where y = [y1, y2,…, yn]. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki−1(yi); yip − yi = Bi, where, as a divisor in Ki−1, Bi has the form (Bi) = qΠpjλj. In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants