Class Field Theory and Applications

Class Field theory deals with the classification of abelian extensions (ie. field extensions with an abelian Galois group). Based on the type of the field we have global CFT (for number field and plane curves over finite fields) as well as local CFT (for $p$-adic fields and Laurent series over finite fields).

Given an extension of number fields $K/k$ a norm equation is trying to find $\alpha \in K$ s.th. $N(\alpha )=\theta$ for a given $\theta \in k$. Classically norm equations are linked to for exmple sums-of-squares: $\theta \in \mathbb{Z}$ is a sum of two squares if $\theta$ is a norm for $\mathbb{Q}(i)$. Norm equations, apart from being classical objects have many applications in algebra.

Classically, the solvability of norm equations is of course investigated locally: if there is a solution, there will ba one modulo every prime. CFT now classfies abelian extensions through suitable norm groups. This can and is used algorithmically, an obstacle being that local solubility is neccessary, but not sufficient, in general.

In this talk I will present some of the core ideas of CFT and their links to norm equations. The corresponding algorithms are practical and available (mostly) through Hecke (hence in Oscar) and also in Magma.