On the generic splitting of quadratic forms in characteristic 2
The theory of generic splitting of quadratic forms goes essentially back to a series of papers by Knebusch in the early to mid 1970s and is at its origin concerned with quadratic forms over fields of characteristic \(\neq 2\) and how they behave after passing from one field to another via places. Of particular interest is the case where the fields involved are extensions of the base field \(F\) over which the form is defined and where the places are \(F\)-places, and function fields of quadratic forms play a central role in this theory. Let \(q\) be a (nonsingular) form over \(F\). The function field \(F(q)\) of \(q\) is the function field of the projective quadric defined by \(q=0\). The isotropy behaviour of \(q\) over field extensions is largely encoded in what is called a generic splitting tower of \(q\) over \(F\), which is defined inductively by \(F_0=F\), \(q_0\) the anisotropic part of \(q\), and for \(i\geq 1\), \(F_i=F_{i-1}(q_{i-1})\) and \(q_i\) the anisotropic part of \(q\) over \(F_i\). By construction, \(\dim q_i<\dim q_{i-1}\), and the smallest \(h\) such that \(\dim q_h\leq 1\) is called the height of \(q\). It turns out that if \(K\) is a field extension of \(F\), then there exists \(i\in\{ 0,\cdots , h\}\) with \(\dim q_i\) equal to the dimension of the anisotropic part of \(q\) over \(K\). Much of this theory goes through also in characteristic \(2\) when considering regular forms (nonsingular forms or forms with nonzero radical of dimension \(1\)) as was shown by \textit{M. Knebusch} and \textit{U. Rehmann} [Contemp. Math. 272, 173-199 (2000; Zbl 0995.11028)]. For arbitrary singular forms, the situation is more complicated and is under current investigation (book in preparation by Knebusch). In the present paper, the author studies the behaviour of (possibly singular) quadratic forms in characteristic \(2\) over the fields in a splitting tower defined as above (which he calls standard splitting tower). If \(q\) is a quadratic form over a field \(F\) of characteristic \(2\), then \(q\cong q'\perp q''\) where \(q'\) is nonsingular and \(q''\) is the restriction of the form to its radical. \(q'\) is not determined uniquely up to isometry, but \(q''\) is, and so is the pair \((r,s)=(\dim q',\dim q'')\) which is called the type of \(q\). The author studies properties of the sequence \((r_i,s_i)\) of types of the forms \(q_i\) as defined above. He shows that an anisotropic \(q\) is of height \(1\) iff \(\dim q=2\), or \(q\) is similar to an \(n\)-fold Pfister form, or \(q\) is the restriction to a subspace of dimension \(2^n-1\) of a form similar to an \(n\)-fold Pfister form. He characterizes so-called excellent forms (whose definition is analogous to that in characteristic \(\neq 2\)). Finally, he gives a classification of nonsingular forms of height \(2\) and degree \(\leq 2\) (i.e. \(\dim q_1=\dim q_{h-1}\in \{ 2,4\}\)).
scientific article; zbMATH DE number 1837067