On isotropy fields of 𝑆𝑝𝑖𝑛(18)-torsors

It is shown that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="18"> <mml:semantics> <mml:mn>18</mml:mn> <mml:annotation encoding="application/x-tex">18</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional nondegenerate quadratic form of trivial discriminant and Clifford invariant acquires Witt index at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="5"> <mml:semantics> <mml:mn>5</mml:mn> <mml:annotation encoding="application/x-tex">5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over some finite ground field extension of degree not divisible by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript 4"> <mml:semantics> <mml:msup> <mml:mn>2</mml:mn> <mml:mn>4</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">2^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. On the basis of previous research, a general formula is also established for all possible similar statements for forms of arbitrary dimension.