WILD GALOIS MODULE STRUCTURE OF RINGS OF INTEGERS AND LEOPOLDT-TYPE RESULTS FOR A4-EXTENSIONS OF Q

Data Seminario: 
26-Sep-2019
Ora Inizio: 
16:00
Ora Fine: 
17:30
Fabio Ferri
University of Exeter
Let $L/K$ be a Galois extension of number fields with Galois group G. We say that the extension has normal integral basis if the ring of integers $\mathcal{O}_L$ is free of rank 1 as an $\mathcal{O}_K[G]$-module; this definition is motivated by the well-known normal basis theorem, which states that in any finite Galois extension of fields $L/K$ with Galois group $G$ we have that $L$ is free of rank 1 over $K[G]$, that is, there is a normal basis of $L$ as a $K$-vector space. A necessary condition for the extension to have normal integral basis is that it is (at most) tamely ramified. If the extension is wildly ramified, we can define an associated order $\mathfrak{A}_{L/K}$ and expect in some situations that $\mathcal{O}_L$ is free of rank 1 as an $\mathfrak{A}_{L/K}$-module. For example, we know from Leopoldt and Bergé that if $K = \mathbb{Q}$ and $G$ is abelian of any order or dihedral of order $2p$, where $p$ is a prime number, then $\mathcal{O}_L$ is always free over $\mathfrak{A}_{L/\mathbb{Q}}$. In this talk we will introduce the theory of wild Galois module structure of rings of integers including the aforementioned theorems. Then we will give a complete characterization of when in an $A_4$-extension $L/\mathbb{Q}$ the ring of integers $\mathcal{O}_L$ is free over its associated order $\mathfrak{A}_{L/\mathbb{Q}}$, which will only depend on the ramification index and inertia degree over the prime 2. This is part of a work in progress concerning Leopoldt-type theorems of nonabelian extensions of $\mathbb{Q}$, which is part of my PhD project with Henri Johnston.