Local Galois module structure in positive characteristic and continued fractions

For a Galois extension of degree p of local fields of characteristic p, we express the Galois action on the ring of integers in terms of a combinatorial object: a balanced {0, 1}-valued sequence that only depends on the discriminant and p. We show that the embedding dimension edim(R) of the associated order R is tightly related to the minimal number d of R-module generators of the ring of integers. Moreover, we show how to compute d and edim(R) from p and the discriminant with a continued fraction expansion. We thank Bruno Anglès, Wieb Bosma and Rob Tijdeman for their bibliographic assistance. Received: 19 March 2006