A note on quadratic forms over Pythagorean fields

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.