On the second trace form of central simple algebras in characteristic two (Q5952933)

If \(A\) is a central simple algebra of degree \(n\) over the field \(K\) and if \(a\in A\) then \(\text{Prd}_A(a)= \det(X\cdot E_n- \varphi(a\otimes L))\in K[X]\) is the reduced characteristic polynomial of \(a\), where \(L\) is a splitting field of \(A\) and \(\varphi: A\otimes L\to M_n(L)\) is an isomorphism onto the matrix algebra. Writing \(\text{Prd}_A(a)= X^n+ s_1(a)\cdot X^{n-1}+ s_2(a)\cdot X^{n-2}+\dots\) one defines the trace form of \(A\), \(T_A:A\to K:a\to s_1(a^2)\), and the second trace form, \(T_{2,A}:A\to K:a\to s_2(a)\). These invariants of the algebra \(A\) have been studied intensively for the case that the characteristic of \(K\) is not 2. When \(\text{char}(K)= 2\) then the trace form has rank 0, whereas the second trace form is nondegenerate if the degree \(n\) is even. Both the Clifford invariant and the Arf invariant of \(A\) are computed for this case.