#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 265-275

** Detlev W. Hoffmann **
Motivic Equivalence
and Similarity of Quadratic Forms

A result by Vishik states that given two anisotropic quadratic forms of
the same dimension over a field of characteristic not $2$, the Chow motives
of the two associated projective quadrics are isomorphic iff both forms
have the same Witt indices over all field extensions, in which case the
two forms are called motivically equivalent. Izhboldin has shown that
if the dimension is odd, then motivic equivalence implies similarity of
the forms. This also holds for even dimension $<= 6$, but Izhboldin
also showed that this generally fails in all even dimensions $\geq 8$
except possibly in dimension $12$. The aim of this paper is to show that
motivic equivalence does imply similarity for fields over which quadratic
forms can be classified by their classical invariants provided that in
the case of formally real such fields the space of orderings has some
nice properties. Examples show that some of the required properties for
the field cannot be weakened.

2010 Mathematics Subject Classification: Primary: 11E04; Secondary: 11E81, 12D15, 14C15

Keywords and Phrases: quadratic form, quadric, function field of a quadric, generic splitting,
similarity, motivic equivalence, formally real field, effective diagonalization

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