Mathematics Colloquium

WHEN:
Thursday, September 27, 2007 at 2:30 PM

WHERE:
Durham Science Center, Room 116

WHAT:

ABSTRACT:
The normal basis theorem is a basic result in abstract algebra that asserts that in a finite, Galois extension of fields (L/K with Galois group G) there is an element a ∈ L such that {g • a: g ∈ G} is a vector space basis for L over K. Fields come in many "flavors". Pick a "flavor" and you can, of course, ask whether anything more can be said.
Our "flavor" will be the class of complete, local fields with finite residue field. These are some of the simplest fields outside of finite fields, and in concrete terms mean that we are talking about fields of power series. They possess a notion of size -- kind of like an absolute value. So it is natural to ask about the size of the element a ∈ L, guaranteed by the normal basis theorem. We will present some recent progress on this question. In general however, there is tension between (1) the ability to easily determine the size of an element and (2) the ability to easily follow the Galois action on a basis for the field extension. And this has presented itself as a major obstruction to progress for the field of local (wildly ramified) Galois module structure.
Remarkably, cyclic extensions of degree p, a prime, do not experience this tension. And so much is known about Galois module structure in these simple extensions. Are there other, equally simple extensions, with much more complicated Galois groups that do not experience this tension? There are. Within these extensions, one can construct a so-called Galois scaffold. We will close with a discussion of recent progress on this topic.

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