Galois scaffolding in characteristic p local extensions
by
Griff Elder
University of Nebraska at Omaha
Coauthors: Nigel Byott

The normal basis theorem says that in a finite Galois extension there is an element whose conjugates serve as a field basis. In the setting of local fields the valuation of an element is a fundamental property. So it is natural ask for a valuation certificate: A valuation that ensures that an element will generate a normal basis.

A necessary condition for a valuation certificate is that the extension be a fully ramified p-extension where the residue characteristic is p. If, in this setting, in addition to a valuation certificate v we ask for a set of t=log_p|G| elements q_i ∈ K[G] such for v_L(r)=v, {v_L(∏_i=1^tq_i^j_ir): 0 ≤ j_i ≤ p-1} is a complete set of residues modulo |G|. Then we have Galois scaffold, which should be considered to be a variant of normal basis especially attuned to valuation.

Galois scaffolds are useful for answering questions in local Galois module structure. In this talk we will discuss criteria that are sufficient for Galois scaffolds and some applications to Galois module structure.

Date received: June 6, 2008