Gauss Genus Theory in Characteristic 2

Abstract

We extend Gauss composition and Gauss genus theory over $\mathbf{Z}$ to $\mathbb{F}_{2^n}[T]$, a polynomial ring over a finite field $\mathbb{F}_{2^n}$ of characteristic 2. We find new invariants of binary quadratic forms over $\mathbb{F}_{2^n}[T]$ by using Arf invariant and introduce new definitions of proper equivalence and direct composition, and prove that the direct composition makes the set of proper equivalence classes of binary quadratic forms with the same invariants into a finite Abelian group, which is isomorphic to a Picard group of a corresponding extension ring of $\mathbb{F}_{2^n}[T]$. Building on this, we develop genus theory in characteristic 2 and prove that the kernel of the generalized Gauss's map is the subgroup of all squares in the class group.