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On quadratic forms over semilocal rings

Using a recent result of Panin and Pimenov we show that several results, as for instance the linkage principle, in the algebraic theory of quadratic forms over fields also hold for quadratic forms over regular semilocal domains which contain a field of characteristic not 2. As an application we prove that the Arason and Elman presentation of the powers of the fundamental ideal of the Witt ring of a field extends to semilocal rings which contain an infinite field of characteristic not 2.