A theorem of T. A. Springer
states that if F is a field of characteristic not two and L is an extension field of F of
odd degree then any anisotropic quadratic form over F remains anisotropic over L. A
weaker version (and an immediate consequence) of this theorem says that the natural
map r : W(F) → W(L), from the Witt ring of F to the Witt ring of L, is injective.
This note investigates the relationship between these statements in the case that L is
a finite Galois extension of a pythagorean field F. Specifically, it is shown that if r is
injective then any anisotropic quadratic form over F remains anisotropic over L
and if, in addition, L is pythagorean then the extension must be of odd
degree. An example is provided of a Galois extension of even degree with r
injective.
