特征2中的Gauss种理论
Gauss Genus Theory in Characteristic 2
AI总结：
将Gauss复合与种理论推广至特征2多项式环，引入新不变量与直接复合，证明类群为有限阿贝尔群且同构于Picard群，并刻画广义Gauss映射的核。
AI中文摘要：
我们将定义在$\mathbf{Z}$上的Gauss复合和Gauss种理论推广到$\mathbb{F}_{2^n}[T]$，即特征2的有限域$\mathbb{F}_{2^n}$上的多项式环。通过使用Arf不变量，我们发现了$\mathbb{F}_{2^n}[T]$上二元二次型的新不变量，并引入了本原等价和直接复合的新定义，证明了直接复合使得具有相同不变量的二元二次型的本原等价类集合构成一个有限阿贝尔群，该群同构于$\mathbb{F}_{2^n}[T]$的相应扩环的Picard群。在此基础上，我们发展了特征2中的种理论，并证明了广义Gauss映射的核是类群中所有平方元素构成的子群。
英文摘要：
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.
