# Henri Johnston

### First Position

Stipendiary lecturer in pure mathematics, St. Hugh's College, Oxford University### Dissertation

*The Trace Map and Galois Module Structure of Rings of Integers for Absolutely Abelian Number Fields*

### Advisor

### Research Area

### Abstract

Let *L*/*K* be a finite Galois extension of number fields with Galois group *G*. Then the group algebra *K*[*G*] operates on the additive structure of *L* in the obvious way and we define the associated order of the extension *L*/*K* to be

A_{L/K} := {*x* ∈ *K*[*G*] | *x*(O_{L}) ⊆ O_{L}},

where O_{L} is the ring of integers of *L*. A natural problem that arises in Galois module theory is that of determining the associated order A_{L/K} and the structure of O_{L} as an A_{L/K}-module.

In the case where *K* = Q and *L*/Q is abelian, this problem was solved by Leopoldt (see [Leo59]; Lettl gives a simplified proof in [Let90]). In particular, Leopoldt showed that O_{L} is a free A_{L/Q}-module (necessarily of rank 1) and gave explicit descriptions of A_{L/Q} and of α ∈ O_{L} such that O_{L }= A_{L/Q} ⋅ α.

This dissertation comes in two main parts, each of which is derived from a paper relating to Leopoldt's Theorem.

The first part is based on [Joh06]. Let *L*/*K* be an extension of number fields where *L*/Q is abelian. Suppose further that *L* and *K* are of equal conductor, *n*. If Tr_{L/K} : *L* → *K* denotes the trace map, then Tr_{L/K}(O_{L}) is an ideal in O_{K}. Sharpening the main result of Girstmair in [Gir92], we determine the norm of Tr_{L/K}(O_{L}) over Q for any such *L*/*K*. The explicit nature of the calculations involved leads to the definition of an "adjusted trace map" that allows the proof of Leopoldt's Theorem to be reduced to the cyclotomic case.

The second part is based on [Joh]. Let *L*/*K* be an extension of number fields where *L*/Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_{L} of *L* is free over the associated order A_{L/K}. Furthermore we define an abelian number field *K* to be Leopoldt if every finite extension *L*/*K* with *L*/Q abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl (see [Leo59], [CL93], [Ble95], and [BL96], respectively) culminate in the proof that the *n*-th cyclotomic field Q^{(n)} is Leopoldt for every *n*. We generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit generators.