Vol. 58, No. 2, 1975

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A note on quadratic forms over Pythagorean fields

Roger P. Ware

Vol. 58 (1975), No. 2, 651–654

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.

Mathematical Subject Classification
Primary: 10C05
Received: 13 March 1974
Published: 1 June 1975
Roger P. Ware