论文标题
特征2中二次和对称双线性形式的相似性
Similarity of quadratic and symmetric bilinear forms in characteristic 2
论文作者
论文摘要
我们说，如果在$ f $上定义任何两种形式（在$ l $上定义的两种形式），则具有等距或对称双线性形式的等轴测或对称双线性形式的等轴测（分别相似性）的下降属性（在$ l $上的$ f $）已经是等轴测（resmetric（resp. spelly）），这已经是等值线（resp。相似（resp.相似）（分别相似）。著名的Artin-Springer定理指出，各向异性二次或对称双线性在野外形成，在奇特的野外扩展上保持各向异性。结果，奇数扩展具有二次和对称双线性等轴测的下降特性。虽然这是以非语言二次形式而闻名的，但它可能以特征性$ 2 $的任意二次或对称双线性形式而闻名。在这种情况下，我们提供了证明。更普遍地，我们表明奇数扩展也具有相似性的下降特性。此外，对于特征性$ 2 $中的对称双线性形式，甚至具有等轴测的下降属性，并且对于任意可分离代数扩展的相似性。我们还向Scharlau的规范原则显示了特征性$ 2 $的任意二次或双线性形式。
We say that a field extension $L/F$ has the descent property for isometry (resp. similarity) of quadratic or symmetric bilinear forms if any two forms defined over $F$ that become isometric (resp. similar) over $L$ are already isometric (resp. similar) over $F$. The famous Artin-Springer theorem states that anisotropic quadratic or symmetric bilinear forms over a field stay anisotropic over an odd degree field extension. As a consequence, odd degree extensions have the descent property for isometry of quadratic as well as symmetric bilinear forms. While this is well known for nonsingular quadratic forms, it is perhaps less well known for arbitrary quadratic or symmetric bilinear forms in characteristic $2$. We provide a proof in this situation. More generally, we show that odd degree extensions also have the descent property for similarity. Moreover, for symmetric bilinear forms in characteristic $2$, one even has the descent property for isometry and for similarity for arbitrary separable algebraic extensions. We also show Scharlau's norm principle for arbitrary quadratic or bilinear forms in characteristic $2$.
