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.

我们说一个字段扩展 $升/F$ 如果任意两个形式定义在 $F$ 成为等距（或相似）超过 $升$ 已经等距（或相似）超过 $F$. 著名的阿廷-斯普林格定理指出，场上的各向异性二次或对称双线性形式在奇数场扩展上保持各向异性。因此，奇次扩展具有二次等距和对称双线性形式的下降特性。虽然这对于非奇异二次型是众所周知的，但对于特征 2 中的任意二次或对称双线性形式可能不太为人所知。我们在这种情况下提供了一个证明。更一般地，我们表明奇数度扩展也具有相似性的下降特性。此外，对于特征 2 中的对称双线性形式，甚至对于任意可分离代数扩展具有等距和相似性的下降特性。