The chain equivalence of totally decomposable orthogonal involutions in characteristic two (Q729746)

A central simple algebra \(A\) over a field \(F\) with an involution of the first kind \(\sigma\) is called ``totally decomposable'' if there exist quaternion algebras \(Q_1,\dots,Q_n\) over \(F\) with involutions \(\sigma_1,\dots,\sigma_n\) such that \((A,\sigma) \cong \otimes_{i=1}^n (Q_i,\sigma_i)\). It is known that if \(\sigma\) is orthogonal then all \(\sigma_i\) are orthogonal too. The main theorem (3.8) states that two totally decomposable algebras \((A,\sigma)=\otimes_{i=1}^n (Q_i,\sigma_i)\) and \((A',\sigma')=\otimes_{i=1}^n (Q_i',\sigma_i')\) with orthogonal involutions over a field \(F\) of \(\text{char}(F)=2\) are isomorphic if and only if they are connected by a chain of isomorphism-class-preserving basic steps modifying either one quaternion algebra \((Q_i,\sigma_i)\) or a pair of quaternion algebras \((Q_i,\sigma_i) \otimes (Q_j,\sigma_j)\) at a time. The principal tools are the bilinear Pfister form (``the Pfister invariant'') associated to a totally decomposable central simple algebra with an orthogonal involution and the chain \(p\)-equivalence for bilinear Pfister forms.